AMC 10 Problems
1300 problems · 2000–2025
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1300 problems · 2000–2025
1300 of 1300 problems
| Description | Topic | |||
|---|---|---|---|---|
| 2025 AMC 10A | Andy and Betsy travel north and east from Mathville; find when they are equidistant from the start. | Algebra | Rates, Mixtures, & Averages | 1 |
| 2025 AMC 10A | Two nut mixes combined to get 40% peanuts; find resulting weight of cashews. | Algebra | Rates, Mixtures, & Averages | 2 |
| 2025 AMC 10A | Count isosceles triangles with integer sides and longest side 2025. | Combinatorics | Counting | 3 |
| 2025 AMC 10A | Students and teachers trivia contest; Ash changes average ages; find Ash's age. | Algebra | Rates, Mixtures, & Averages | 4 |
| 2025 AMC 10A | Find the 2025th term of the sequence 1,2,1,2,3,2,1,... | Algebra | Sequences & Series | 5 |
| 2025 AMC 10A | Equilateral triangle trisectors form a convex hexagon; find its smallest angle. | Geometry | Triangles & Polygons | 6 |
| 2025 AMC 10A | Polynomial f(x)=x³+x²+ax+b; given remainders for x-1 and x-2, find b-a. | Algebra | Polynomials | 7 |
| 2025 AMC 10A | Agnes writes four self-referential statements; determine the number of false ones. | Algebra | Logic & Truth Values | 8 |
| 2025 AMC 10A | Given f(x)=100x³-300x²+200x, count a such that y=f(x-a) passes through (1,25). | Algebra | Polynomials | 9 |
| 2025 AMC 10A | Semicircle with chord CD of length 16 parallel to diameter AB; smaller semicircle removed; find shaded area. | Geometry | Circular Geometry | 10 |
| 2025 AMC 10A | Arithmetic sequence 1,x,y,z and geometric 1,p,q,z; z minimal integer; find x+y+z+p+q. | Algebra | Sequences & Series | 11 |
| 2025 AMC 10A | 4-digit passcode with one even digit, one prime digit, no zero; count possible codes. | Combinatorics | Counting | 12 |
| 2025 AMC 10A | Infinitely nested squares alternately shaded; shaded area is 64% of original; find reduction ratio k. | Algebra | Sequences & Series | 13 |
| 2025 AMC 10A | Six chairs around a round table; 2 students and 2 teachers randomly sit; probability they sit adjacent in pairs. | Combinatorics | Probability | 14 |
| 2025 AMC 10A | Rectangle ABEF with lengths 7,1,5; AD perpendicular to DE; find area of triangle ABC. | Geometry | Triangles & Polygons | 15 |
| 2025 AMC 10A | Three coins randomly placed into three jars; expected number in the jar with the most coins. | Combinatorics | Probability | 16 |
| 2025 AMC 10A | N is the unique positive integer with 273436 mod N = 16 and 272760 mod N = 15; find tens digit of N. | Number Theory | Factors, Multiples & Divisibility | 17 |
| 2025 AMC 10A | Harmonic mean of all real roots of the product of 2025 quadratics of form kx²-4x-3. | Algebra | Polynomials | 18 |
| 2025 AMC 10A | Array built from -1,3,1 with adjacent sums; one row sums to 12288; find its third number. | Algebra | Sequences & Series | 19 |
| 2025 AMC 10A | Silo (diameter 20); MacDonald and McGregor positioned so line of sight is tangent; find distance g. | Geometry | Circular Geometry | 20 |
| 2025 AMC 10A | Find the maximum size of a sum-free subset of {1,2,...,20}. | Combinatorics | Counting | 21 |
| 2025 AMC 10A | Circle of radius r surrounded by three circles of radii 1,2,3 all externally tangent; find r. | Geometry | Circular Geometry | 22 |
| 2025 AMC 10A | Triangle ABC sides 80,45,75; angle bisector of B and altitude to AB intersect at P; find BP. | Geometry | Triangles & Polygons | 23 |
| 2025 AMC 10A | Count fair integers: distinct nonzero digits, no digit adjacent to two larger digits. | Combinatorics | Counting | 24 |
| 2025 AMC 10A | Point P inside square ABCD; probability AP is the middle length among AP, BP, AB. | Combinatorics | Probability | 25 |
| 2025 AMC 10B | Max mugs from 350g bag at 20g per mug | Algebra | Rates, Mixtures, & Averages | 1 |
| 2025 AMC 10B | Sum of ones digits of squares 1^2 to 2025^2 | Number Theory | Modular Arithmetic | 2 |
| 2025 AMC 10B | Sum of digits of sum of 11th row in Pascal-like triangle | Algebra | Sequences & Series | 3 |
| 2025 AMC 10B | Two-digit base equality: ab (base7) = ba (base9), find sum a+b | Algebra | Systems & Algebraic Manipulation | 4 |
| 2025 AMC 10B | Angle CAO in triangle with given sides and angle B=130 | Geometry | Circular Geometry | 5 |
| 2025 AMC 10B | Area division of square by line, find equal-area line x=a | Geometry | Coordinate Geometry | 6 |
| 2025 AMC 10B | Distance paths to box via unlocked gates, equal distances | Geometry | Coordinate Geometry | 7 |
| 2025 AMC 10B | Cost $ABBA divisible by 36, find digit sum A+B | Number Theory | Factors, Multiples & Divisibility | 8 |
| 2025 AMC 10B | Count integer solutions to system of inequalities | Algebra | Inequalities | 9 |
| 2025 AMC 10B | Integers n such that rational function f(n)/g(n) is integer | Algebra | Polynomials | 10 |
| 2025 AMC 10B | Probability exactly 2 students same tutor both days | Combinatorics | Probability | 11 |
| 2025 AMC 10B | Compare area ratios of disks to enclosing polygons | Geometry | Triangles & Polygons | 12 |
| 2025 AMC 10B | Ratio of segments on altitude in 30-60-90 triangle | Geometry | Triangles & Polygons | 13 |
| 2025 AMC 10B | Probability tallest 3 athletes are group captains | Combinatorics | Probability | 14 |
| 2025 AMC 10B | Infinite sum of rational series in k | Algebra | Sequences & Series | 15 |
| 2025 AMC 10B | Distinct colorings of 6-sector circle with 2 of each color, no adjacent same | Combinatorics | Counting | 16 |
| 2025 AMC 10B | Max n for decreasing integer sequence with given average drop | Algebra | Sequences & Series | 17 |
| 2025 AMC 10B | Ones digit of sum of floor(√n) for n=1 to 2025 | Number Theory | Modular Arithmetic | 18 |
| 2025 AMC 10B | Fill time for remainder of container with trapezoidal sides | Geometry | 3D Geometry & Solids | 19 |
| 2025 AMC 10B | Diameter of circle tangent to four inscribed semicircles in square | Geometry | Circular Geometry | 20 |
| 2025 AMC 10B | Distinct colorings of 3x3 grid with cyclic adjacency constraint | Combinatorics | Counting | 21 |
| 2025 AMC 10B | Probability random 7-digit number with digit sum 61 is divisible by 11 | Combinatorics | Probability | 22 |
| 2025 AMC 10B | Count squares with same number in two fill orders of 141x91 grid | Number Theory | Factors, Multiples & Divisibility | 23 |
| 2025 AMC 10B | Probability frog reaches 4 with given hop and disappear probabilities | Combinatorics | Probability | 24 |
| 2025 AMC 10B | Path from P to Q reflecting in square, find terminal vertex | Geometry | Coordinate Geometry | 25 |
| 2024 AMC 10A | Evaluate 9901·101 − 99·10101. | Algebra | Systems & Algebraic Manipulation | 1 |
| 2024 AMC 10A | Hiking time model T=aL+bG; find T for given L and G. | Algebra | Systems & Algebraic Manipulation | 2 |
| 2024 AMC 10A | Least prime that is sum of 5 distinct primes; sum its digits. | Number Theory | Factors, Multiples & Divisibility | 3 |
| 2024 AMC 10A | 2024 as sum of two-digit numbers, least number of terms needed. | Algebra | Systems & Algebraic Manipulation | 4 |
| 2024 AMC 10A | Least n such that n! is a multiple of 2024. | Number Theory | Factors, Multiples & Divisibility | 5 |
| 2024 AMC 10A | Minimum adjacent swaps to reverse string ABCDEF. | Combinatorics | Counting | 6 |
| 2024 AMC 10A | Three integers multiply to 60, least possible positive sum. | Number Theory | Factors, Multiples & Divisibility | 7 |
| 2024 AMC 10A | Chocolate factory packing rates; find when Daria joined. | Algebra | Rates, Mixtures, & Averages | 8 |
| 2024 AMC 10A | 6 juniors, 6 seniors into 3 teams of 2 each, count ways. | Combinatorics | Counting | 9 |
| 2024 AMC 10A | Apply operation (divide by 3 or add 10) 100 times starting at 100. | Algebra | Sequences & Series | 10 |
| 2024 AMC 10A | Integer solutions to sqrt(n²−49)=m; count ordered pairs. | Number Theory | Factors, Multiples & Divisibility | 11 |
| 2024 AMC 10A | Zelda's daily score changes over 6 days; find average score. | Algebra | Rates, Mixtures, & Averages | 12 |
| 2024 AMC 10A | Which pairs of transformations (translate, rotate, reflect, dilate) commute? | Combinatorics | Counting | 13 |
| 2024 AMC 10A | Equilateral triangle, tangent circle; area of exterior region. | Geometry | Circular Geometry | 14 |
| 2024 AMC 10A | M+1213 and M+3773 are perfect squares; units digit of M. | Number Theory | Factors, Multiples & Divisibility | 15 |
| 2024 AMC 10A | Nested similar rectangles with given areas; find length AB. | Geometry | Triangles & Polygons | 16 |
| 2024 AMC 10A | Best-of-3 playoff probabilities; find away win probability p. | Combinatorics | Probability | 17 |
| 2024 AMC 10A | Bases b where 2024_b is divisible by 16; count and sum digits of count. | Number Theory | Modular Arithmetic | 18 |
| 2024 AMC 10A | Geometric sequence a, 720, b; least b with sum of digits. | Algebra | Sequences & Series | 19 |
| 2024 AMC 10A | Subset of 1..2024 with spacing constraints; max possible size. | Combinatorics | Counting | 20 |
| 2024 AMC 10A | 5×5 grid with arithmetic rows/columns; find missing entry. | Algebra | Sequences & Series | 21 |
| 2024 AMC 10A | Eight kites made of 30-60-90 triangles form polygon; area of triangle ABC. | Geometry | Triangles & Polygons | 22 |
| 2024 AMC 10A | Integers a,b,c satisfy three equations; find ab+bc+ca. | Algebra | Systems & Algebraic Manipulation | 23 |
| 2024 AMC 10A | Bee moves on 3D grid via die rolls; probability of traversing four distinct edges of a unit cube. | Combinatorics | Probability | 24 |
| 2024 AMC 10A | Place toothpicks on 8×3 grid to form closed loop; count ways. | Combinatorics | Counting | 25 |
| 2024 AMC 10B | 1013th from left is 1010th from right; find total number of people in line | Algebra | Systems & Algebraic Manipulation | 1 |
| 2024 AMC 10B | Evaluate 10! - 7! * 6! | Algebra | Systems & Algebraic Manipulation | 2 |
| 2024 AMC 10B | Count integer x with |2x| ≤ 7π | Algebra | Inequalities | 3 |
| 2024 AMC 10B | Balls placed in bins A-E cyclically with increasing count; find bin for ball 2024 | Algebra | Sequences & Series | 4 |
| 2024 AMC 10B | Change + to – in sum of odds to make total negative; least number of changes | Algebra | Sequences & Series | 5 |
| 2024 AMC 10B | Rectangle integer sides area 2024; least possible perimeter | Number Theory | Factors, Multiples & Divisibility | 6 |
| 2024 AMC 10B | Remainder of 7^{2024}+7^{2025}+7^{2026} divided by 19 | Number Theory | Modular Arithmetic | 7 |
| 2024 AMC 10B | Product of all positive divisors of 42; find units digit | Number Theory | Factors, Multiples & Divisibility | 8 |
| 2024 AMC 10B | Arithmetic mean of a,b,c is 0, of squares is 10; find mean of ab,ac,bc | Algebra | Systems & Algebraic Manipulation | 9 |
| 2024 AMC 10B | Parallelogram with midpoint E; intersection F of EB and AC; area ratio of CDEF to CFB | Geometry | Triangles & Polygons | 10 |
| 2024 AMC 10B | Rectangle with points M, A, right angle, equal triangle areas; find area of WMA | Geometry | Triangles & Polygons | 11 |
| 2024 AMC 10B | 100 students, each speaks k languages, any pair lacks some common language; least total languages | Combinatorics | Counting | 12 |
| 2024 AMC 10B | √x+√y=√1183; positive integers x,y; minimize x+y | Algebra | Inequalities | 13 |
| 2024 AMC 10B | Probability dart in region (x²+y²-25)²≤49 inside |x|+|y|≤8 | Geometry | Coordinate Geometry | 14 |
| 2024 AMC 10B | List of 9 numbers, given six, x≤y≤z, range 7, integer mean and median; count triples (x,y,z) | Algebra | Systems & Algebraic Manipulation | 15 |
| 2024 AMC 10B | Repeatedly replace four numbers by sum/product; all remaining numbers odd; max possible count | Number Theory | Modular Arithmetic | 16 |
| 2024 AMC 10B | 5 snails race, at most one tie (any size); number of possible results | Combinatorics | Counting | 17 |
| 2024 AMC 10B | Possible remainders when an integer to the 100th power is divided by 125 | Number Theory | Modular Arithmetic | 18 |
| 2024 AMC 10B | For slopes 0, rational nonzero, irrational; which numbers of lattice points are possible | Geometry | Coordinate Geometry | 19 |
| 2024 AMC 10B | Three pairs of shoes; no left shoe next to right shoe from a different pair; number of arrangements | Combinatorics | Counting | 20 |
| 2024 AMC 10B | Two pipes radii 1 and 1/4 on floor; third pipe tangent to both and floor; sum of possible radii | Geometry | Circular Geometry | 21 |
| 2024 AMC 10B | 16 people into 4 indistinguishable committees of 4, each with chair and secretary; count assignments, find exponent of 3 | Combinatorics | Counting | 22 |
| 2024 AMC 10B | Sum of F₂/F₁ + F₄/F₂ + ... + F₂₀/F₁₀ for Fibonacci numbers | Algebra | Sequences & Series | 23 |
| 2024 AMC 10B | P(m)=m/2 + m²/4 + m⁴/8 + m⁸/8; how many of P(2022..2025) are integers | Number Theory | Modular Arithmetic | 24 |
| 2024 AMC 10B | Bricks a×b×c form 3×3×3 block; reconfigured to 2×2×7 block 1 unit larger each dimension; find a+b+c | Algebra | Systems & Algebraic Manipulation | 25 |
| 2023 AMC 10A | Alicia and Beth bike toward each other from cities 45 miles apart; find meeting distance from city A. | Algebra | Rates, Mixtures, & Averages | 1 |
| 2023 AMC 10A | Weight of a large pizza given a relationship with orange slices and known slice weight. | Algebra | Systems & Algebraic Manipulation | 2 |
| 2023 AMC 10A | Count positive perfect squares less than 2023 that are divisible by 5. | Number Theory | Factors, Multiples & Divisibility | 3 |
| 2023 AMC 10A | Greatest possible side length of a quadrilateral with integer sides, perimeter 26, and one side 4. | Geometry | Triangles & Polygons | 4 |
| 2023 AMC 10A | How many digits are in the base‑ten representation of 8⁵·5¹⁰·15⁵? | Algebra | Logarithms & Exponents | 5 |
| 2023 AMC 10A | Value of a cube defined by vertex, edge, and face sums given the vertex sum is 21. | Combinatorics | Counting | 6 |
| 2023 AMC 10A | Probability that a running total of 4 die rolls equals 3 at some point. | Combinatorics | Probability | 7 |
| 2023 AMC 10A | Convert 200°F to the linear Breadus scale given two reference points. | Algebra | Rates, Mixtures, & Averages | 8 |
| 2023 AMC 10A | For 2023, count dates YYYYMMDD where each digit appears an even number of times. | Combinatorics | Counting | 9 |
| 2023 AMC 10A | Maureen’s current quiz mean given how extra scores change the average. | Algebra | Rates, Mixtures, & Averages | 10 |
| 2023 AMC 10A | Ratio of shorter to longer leg in a shaded right triangle between two squares of areas 2 and 3. | Geometry | Triangles & Polygons | 11 |
| 2023 AMC 10A | Number of three‑digit multiples of 7 whose digit reversal is a multiple of 5. | Number Theory | Factors, Multiples & Divisibility | 12 |
| 2023 AMC 10A | Square of the maximum distance from Abdul to Bharat given fixed angle and distance between Abdul and Chiang. | Geometry | Triangles & Polygons | 13 |
| 2023 AMC 10A | Probability that a randomly chosen divisor of a random integer 1–100 is divisible by 11. | Combinatorics | Probability | 14 |
| 2023 AMC 10A | Least even number of nested circles needed for total shaded area ≥ 2023π. | Algebra | Sequences & Series | 15 |
| 2023 AMC 10A | Total games played in a table tennis tournament given a player handedness ratio and a win ratio. | Algebra | Systems & Algebraic Manipulation | 16 |
| 2023 AMC 10A | Perimeter of triangle APQ in a rectangle with integer side lengths for three right triangles. | Geometry | Triangles & Polygons | 17 |
| 2023 AMC 10A | How many vertices of a rhombic dodecahedron have exactly 3 edges meeting? | Geometry | 3D Geometry & Solids | 18 |
| 2023 AMC 10A | Center of rotation P(r,s) for a segment rotated to another; find |r−s|. | Geometry | Coordinate Geometry | 19 |
| 2023 AMC 10A | Number of colorings of a 3×3 grid with four colors such that every 2×2 block has all four colors. | Combinatorics | Counting | 20 |
| 2023 AMC 10A | Find m+n where the unique non‑integer root of a minimal polynomial is m/n. | Algebra | Polynomials | 21 |
| 2023 AMC 10A | Radius of the fourth circle internally tangent to two unit circles and externally tangent to a larger inscribed circle. | Geometry | Circular Geometry | 22 |
| 2023 AMC 10A | Sum of digits of N where N has two pairs of complementary divisors with differences 20 and 23. | Number Theory | Factors, Multiples & Divisibility | 23 |
| 2023 AMC 10A | Area inside a large regular hexagon not covered by six smaller regular hexagons, given a spacing of 3/7. | Geometry | Triangles & Polygons | 24 |
| 2023 AMC 10A | Probability that graph distance on a regular icosahedron satisfies d(Q,R) > d(R,S) for three random distinct vertices. | Combinatorics | Probability | 25 |
| 2023 AMC 10B | Equalizing orange juice amounts among four glasses by pouring. | Algebra | Rates, Mixtures, & Averages | 1 |
| 2023 AMC 10B | Find original price of shoes given discount, tax, and total money. | Algebra | Rates, Mixtures, & Averages | 2 |
| 2023 AMC 10B | Ratio of areas of circles circumscribing 3-4-5 and 5-12-13 right triangles. | Geometry | Circular Geometry | 3 |
| 2023 AMC 10B | Area covered by paint strip given width in mm, length in m, unit conversion. | Algebra | Rates, Mixtures, & Averages | 4 |
| 2023 AMC 10B | Find number of terms in list given sums after adding 3 and multiplying by 3. | Algebra | Systems & Algebraic Manipulation | 5 |
| 2023 AMC 10B | Number of even terms in recurrence L1=1, L2=3, L_{n+2}=L_{n+1}+L_n. | Algebra | Sequences & Series | 6 |
| 2023 AMC 10B | Angle between side of square and its copy rotated 20° about center. | Geometry | Triangles & Polygons | 7 |
| 2023 AMC 10B | Units digit of 2022^2023 + 2023^2022. | Number Theory | Modular Arithmetic | 8 |
| 2023 AMC 10B | Number of consecutive perfect square pairs with difference at most 2023. | Algebra | Sequences & Series | 9 |
| 2023 AMC 10B | Minimum guesses to guarantee finding a hidden 2x1 rectangle on 3x3 grid. | Combinatorics | Counting | 10 |
| 2023 AMC 10B | Number of bill combinations ($20, $50, $100) totaling $800 with each denomination used. | Combinatorics | Counting | 11 |
| 2023 AMC 10B | Number of intervals where P(x) is positive after removing its 10 roots. | Algebra | Polynomials | 12 |
| 2023 AMC 10B | Area of region defined by ||x|-1|+||y|-1| ≤ 1. | Geometry | Coordinate Geometry | 13 |
| 2023 AMC 10B | Integer solutions (m,n) to m^2+mn+n^2 = m^2 n^2. | Algebra | Systems & Algebraic Manipulation | 14 |
| 2023 AMC 10B | Smallest m making m·2!·3!·...·16! a perfect square. | Number Theory | Factors, Multiples & Divisibility | 15 |
| 2023 AMC 10B | Difference between number of upnos and downnos of at least two digits. | Combinatorics | Counting | 16 |
| 2023 AMC 10B | Longest interior diagonal of box given edge sum, face area sum, volume. | Algebra | Systems & Algebraic Manipulation | 17 |
| 2023 AMC 10B | Which gcd statements are true given a/14 + b/15 = c/210? | Number Theory | Factors, Multiples & Divisibility | 18 |
| 2023 AMC 10B | Probability a frog jumping randomly from a point in a square lands outside. | Combinatorics | Probability | 19 |
| 2023 AMC 10B | Length of curve of four congruent semicircles drawn on a sphere of radius 2. | Geometry | 3D Geometry & Solids | 20 |
| 2023 AMC 10B | Probability 2023 balls placed randomly into 3 bins all have odd counts. | Combinatorics | Probability | 21 |
| 2023 AMC 10B | Number of solutions to floor(x)^2 - 3x + 2 = 0. | Algebra | Functions | 22 |
| 2023 AMC 10B | Find a+d+n for arithmetic sequence with sum off by 1 from 222. | Algebra | Sequences & Series | 23 |
| 2023 AMC 10B | Perimeter of parametric region (2u-3w, v+4w) for u,v,w in [0,1]. | Geometry | Coordinate Geometry | 24 |
| 2023 AMC 10B | Area of smaller pentagon formed by folding vertices of a regular pentagon to center. | Geometry | Triangles & Polygons | 25 |
| 2022 AMC 10A | Evaluate the continued fraction 3 + 1/(3 + 1/(3 + 1/3)). | Algebra | Systems & Algebraic Manipulation | 1 |
| 2022 AMC 10A | Mike cycles 15 laps in 57 min; estimate laps in first 27 min. | Algebra | Rates, Mixtures, & Averages | 2 |
| 2022 AMC 10A | Find absolute difference of first and second numbers given sum and relationships. | Algebra | Systems & Algebraic Manipulation | 3 |
| 2022 AMC 10A | Convert fuel efficiency from miles per gallon to liters per 100 km with general units. | Algebra | Rates, Mixtures, & Averages | 4 |
| 2022 AMC 10A | Square side 1, equilateral convex hexagon inscribed, find side length s. | Geometry | Triangles & Polygons | 5 |
| 2022 AMC 10A | Simplify |a - 2 - √((a-1)^2)| for a < 0. | Algebra | Functions | 6 |
| 2022 AMC 10A | Find sum of digits of n given LCM(n,18)=180 and GCD(n,45)=15. | Number Theory | Factors, Multiples & Divisibility | 7 |
| 2022 AMC 10A | Mean of six numbers equals a value in set; find sum of possible X. | Algebra | Rates, Mixtures, & Averages | 8 |
| 2022 AMC 10A | Color 5 regions of a partitioned rectangle with 5 colors, adjacent differ. | Combinatorics | Counting | 9 |
| 2022 AMC 10A | Rectangle with diagonal 8, cut 1x1 squares, distance 4√2, find area. | Geometry | Triangles & Polygons | 10 |
| 2022 AMC 10A | Equation 2^m √(1/4096) = 2 √[m](1/4096); find sum of real m. | Algebra | Logarithms & Exponents | 11 |
| 2022 AMC 10A | Truth-tellers, liars, alternaters answer questions; candy given; find truth-tellers' candy. | Algebra | Logic & Truth Values | 12 |
| 2022 AMC 10A | Triangle with angle bisector AP, perpendicular BD, AD parallel BC; find AD. | Geometry | Triangles & Polygons | 13 |
| 2022 AMC 10A | Split integers 1-14 into 7 pairs where greater ≥ 2× lesser; count ways. | Combinatorics | Counting | 14 |
| 2022 AMC 10A | Area between circle and inscribed quadrilateral with sides 7,24,20,15. | Geometry | Circular Geometry | 15 |
| 2022 AMC 10A | Volume after lengthening each edge of box by 2; roots of 10x^3-39x^2+29x-6. | Algebra | Polynomials | 16 |
| 2022 AMC 10A | Three-digit with repeating decimal equation; count integer solutions. | Algebra | Systems & Algebraic Manipulation | 17 |
| 2022 AMC 10A | Sequence of k° rotations and y-axis reflections returns (1,0); find least n. | Geometry | Coordinate Geometry | 18 |
| 2022 AMC 10A | Sum of reciprocals 1 to 17 equals h/L_17; find h mod 17. | Number Theory | Modular Arithmetic | 19 |
| 2022 AMC 10A | Fourth term of sum of arithmetic and geometric sequences given first three sums. | Algebra | Sequences & Series | 20 |
| 2022 AMC 10A | Bowl of hexagons on square; find area of top octagon. | Geometry | Triangles & Polygons | 21 |
| 2022 AMC 10A | Cards 1-13 arranged; picked up in two passes; count orderings. | Combinatorics | Counting | 22 |
| 2022 AMC 10A | Isosceles trapezoid ABCD, point P with distances 1,2,3,4; find BC/AD. | Geometry | Triangles & Polygons | 23 |
| 2022 AMC 10A | Strings of digits 0-4 length 5 with at least j digits < j; count strings. | Combinatorics | Counting | 24 |
| 2022 AMC 10A | Squares R,S,T with lattice points conditions; find sum of edge lengths. | Algebra | Systems & Algebraic Manipulation | 25 |
| 2022 AMC 10B | Evaluate expression with custom absolute difference operation. | Algebra | Functions | 1 |
| 2022 AMC 10B | Find area of rhombus given PB perpendicular to AD, AP=3, PD=2. | Geometry | Triangles & Polygons | 2 |
| 2022 AMC 10B | Count three-digit numbers with an odd number of even digits. | Combinatorics | Counting | 3 |
| 2022 AMC 10B | Find time of 700th hiccup starting at 4:00, occurring every 5 seconds. | Algebra | Rates, Mixtures, & Averages | 4 |
| 2022 AMC 10B | Compute fraction involving products of fractions and a square root. | Algebra | Systems & Algebraic Manipulation | 5 |
| 2022 AMC 10B | Determine how many of first ten terms of a specific sequence are prime. | Number Theory | Factors, Multiples & Divisibility | 6 |
| 2022 AMC 10B | Find number of k such that x^2+kx+36 has two distinct integer roots. | Algebra | Polynomials | 7 |
| 2022 AMC 10B | Find how many of 100 sets of 10 consecutive integers contain exactly two multiples of 7. | Number Theory | Factors, Multiples & Divisibility | 8 |
| 2022 AMC 10B | Sum the series of fractions n/(n+1)! from 1 to 2021. | Algebra | Sequences & Series | 9 |
| 2022 AMC 10B | Five positive integers with unique mode 2 more than median, median 2 more than mean; find least mode. | Algebra | Rates, Mixtures, & Averages | 10 |
| 2022 AMC 10B | Logical equivalence of 'No school bigger than Euclid sold more T-shirts.' | Algebra | Logic & Truth Values | 11 |
| 2022 AMC 10B | Least n such that probability of rolling sum 7 at least once in n rolls > 1/2. | Combinatorics | Probability | 12 |
| 2022 AMC 10B | Two primes differ by 2, difference of cubes is 31106; find sum of digits of next prime. | Algebra | Systems & Algebraic Manipulation | 13 |
| 2022 AMC 10B | Maximum size of subset of 1..25 where sum of any two elements is never in the set. | Combinatorics | Counting | 14 |
| 2022 AMC 10B | Given arithmetic sequence (d=2), S_{3n}/S_n constant; find S_20. | Algebra | Sequences & Series | 15 |
| 2022 AMC 10B | Rectangle 4×8, square side 5 placed on it; find area of overlap. | Geometry | Triangles & Polygons | 16 |
| 2022 AMC 10B | Which number among five given is not divisible by any prime less than 10. | Number Theory | Modular Arithmetic | 17 |
| 2022 AMC 10B | How many systems of three linear equations with 0/1 coefficients have a nonzero solution. | Combinatorics | Counting | 18 |
| 2022 AMC 10B | Game of Life on 5×5 board; count initial configurations yielding center filled after one step. | Combinatorics | Counting | 19 |
| 2022 AMC 10B | Rhombus with angle 46°, midpoint, perpendicular; find angle BFC. | Geometry | Triangles & Polygons | 20 |
| 2022 AMC 10B | Find minimal-degree polynomial P(x) given remainders modulo x^2+x+1 and x^2+1; sum of squares of coefficients. | Algebra | Polynomials | 21 |
| 2022 AMC 10B | Sum of areas of all circles tangent to three given circles in the plane. | Geometry | Circular Geometry | 22 |
| 2022 AMC 10B | Ant Amelia: random increments and times uniform (0,1), up to 3 steps; probability position > 1. | Combinatorics | Probability | 23 |
| 2022 AMC 10B | Given Lipschitz condition and f(300)=f(900), maximize |f(f(800))-f(f(400))|. | Algebra | Functions | 24 |
| 2022 AMC 10B | Binary sequence x_k such that sum S_n satisfies 7S_n ≡ 1 mod 2^n; find weighted sum of four terms. | Number Theory | Modular Arithmetic | 25 |
| 2021 Fall AMC 10A | Evaluate the expression (2112-2021)^2 divided by 169. | Algebra | Systems & Algebraic Manipulation | 1 |
| 2021 Spring AMC 10A | Evaluate the expression (2^2-2)-(3^2-3)+(4^2-4). | Algebra | Systems & Algebraic Manipulation | 1 |
| 2021 Fall AMC 10A | Find new area after shortening the other side of a 4x6 card by 1 inch when shortening one side gave area 18. | Algebra | Systems & Algebraic Manipulation | 2 |
| 2021 Spring AMC 10A | Portia's school has three times as many students as Lara's; total 2600. Find Portia's students. | Algebra | Rates, Mixtures, & Averages | 2 |
| 2021 Fall AMC 10A | Maximum number of reshapable clay balls of radius 2 that fit in a cube of side 6. | Geometry | 3D Geometry & Solids | 3 |
| 2021 Spring AMC 10A | Sum of two natural numbers is 17,402; one divisible by 10, erasing units digit gives other. Find difference. | Algebra | Systems & Algebraic Manipulation | 3 |
| 2021 Fall AMC 10A | How many minutes quicker is Route B than Route A given speeds and a school zone? | Algebra | Rates, Mixtures, & Averages | 4 |
| 2021 Spring AMC 10A | Cart accelerates down hill; distance each second increases by 7 inches. Find total distance in 30 seconds. | Algebra | Sequences & Series | 4 |
| 2021 Fall AMC 10A | Find the digit A such that 20210A is prime. | Number Theory | Factors, Multiples & Divisibility | 5 |
| 2021 Spring AMC 10A | Class mean of k>12 students is 8; 12 have mean 14. Find mean of remaining in terms of k. | Algebra | Rates, Mixtures, & Averages | 5 |
| 2021 Fall AMC 10A | Oscar's leap vs Elmer's stride length given strides between poles and total distance. | Algebra | Rates, Mixtures, & Averages | 6 |
| 2021 Spring AMC 10A | Chantal and Jean hike; given speeds, find Jean's average speed until they meet. | Algebra | Rates, Mixtures, & Averages | 6 |
| 2021 Fall AMC 10A | In a square with point E outside, find angle AFE given angle CDE = 110° and DE = DF. | Geometry | Triangles & Polygons | 7 |
| 2021 Spring AMC 10A | Given statements about snakes (happy, can add, purple, can't subtract), which conclusion is valid? | Algebra | Logic & Truth Values | 7 |
| 2021 Fall AMC 10A | Count two-digit cuddly numbers equal to tens digit plus units digit squared. | Algebra | Systems & Algebraic Manipulation | 8 |
| 2021 Spring AMC 10A | Multiplying 66 by a repeating decimal incorrectly gives answer 0.5 less. Find the two-digit repeating part. | Algebra | Systems & Algebraic Manipulation | 8 |
| 2021 Fall AMC 10A | Probability sum is even when rolling an unfair die twice (even 3x as likely as odd). | Combinatorics | Probability | 9 |
| 2021 Spring AMC 10A | Find the least possible value of (xy-1)^2 + (x+y)^2 for real x and y. | Algebra | Inequalities | 9 |
| 2021 Fall AMC 10A | t = average class size from teacher's view, s = from student's view; find t - s. | Algebra | Rates, Mixtures, & Averages | 10 |
| 2021 Spring AMC 10A | Which expression equals the product (2+3)(2^2+3^2)…(2^{64}+3^{64})? | Algebra | Polynomials | 10 |
| 2021 Fall AMC 10A | Emily walks past a ship, counts steps; find length of ship in her steps. | Algebra | Rates, Mixtures, & Averages | 11 |
| 2021 Spring AMC 10A | For which base b is 2021_b − 221_b not divisible by 3? | Number Theory | Modular Arithmetic | 11 |
| 2021 Fall AMC 10A | Remainder when base-nine number 27,006,000,052_nine is divided by 5. | Number Theory | Modular Arithmetic | 12 |
| 2021 Spring AMC 10A | Two cones with equal liquid volume; marble dropped. Find ratio of liquid level rises. | Geometry | 3D Geometry & Solids | 12 |
| 2021 Fall AMC 10A | Probability that each of 6 balls differs in color from more than half the others. | Combinatorics | Probability | 13 |
| 2021 Spring AMC 10A | Find volume of tetrahedron with given edge lengths. | Geometry | 3D Geometry & Solids | 13 |
| 2021 Fall AMC 10A | Number of ordered real pairs satisfying x^2+3y=9 and (|x|+|y|-4)^2=1. | Algebra | Systems & Algebraic Manipulation | 14 |
| 2021 Spring AMC 10A | Polynomial with positive integer roots has given form; find the coefficient B. | Algebra | Polynomials | 14 |
| 2021 Fall AMC 10A | Area of circle through vertices A,B,C of isosceles triangle with tangent circle. | Geometry | Circular Geometry | 15 |
| 2021 Spring AMC 10A | Select distinct A,B,C,D from 1-6; two parabolas intersect. How many selections? | Combinatorics | Counting | 15 |
| 2021 Fall AMC 10A | Find the symmetry of f(x) = |floor(x)| - |floor(1-x)|. | Algebra | Functions | 16 |
| 2021 Spring AMC 10A | List where integer n appears n times up to 200. Find the median of the list. | Algebra | Sequences & Series | 16 |
| 2021 Fall AMC 10A | Height of pillar at E in a regular hexagon given heights at A, B, C support a flat solar panel. | Geometry | 3D Geometry & Solids | 17 |
| 2021 Spring AMC 10A | Trapezoid with given side lengths and right angle; find AD length. | Geometry | Triangles & Polygons | 17 |
| 2021 Fall AMC 10A | Number of ways to plant four crops in a 2x2 grid avoiding certain adjacent pairs. | Combinatorics | Counting | 18 |
| 2021 Spring AMC 10A | Function f on rationals with f(ab)=f(a)+f(b) and f(p)=p for prime p. For which x is f(x)<0? | Algebra | Functions | 18 |
| 2021 Fall AMC 10A | Side length s of square given swept areas of a disk rolling inside and outside. | Geometry | Circular Geometry | 19 |
| 2021 Spring AMC 10A | Find area of region bounded by x^2+y^2 = 3|x-y|+3|x+y|. | Geometry | Coordinate Geometry | 19 |
| 2021 Fall AMC 10A | Count positive integer pairs (b,c) so neither quadratic has two distinct real roots. | Algebra | Polynomials | 20 |
| 2021 Spring AMC 10A | Number of rearrangements of 1..5 with no three consecutive increasing or decreasing terms. | Combinatorics | Counting | 20 |
| 2021 Fall AMC 10A | Ratio p/q where p is probability of specific bin counts (3,5,4,4,4) and q of (4,4,4,4,4) when tossing 20 balls into 5 bins. | Combinatorics | Probability | 21 |
| 2021 Spring AMC 10A | Equiangular hexagon; given areas of triangles formed by extending sides, find perimeter. | Geometry | Triangles & Polygons | 21 |
| 2021 Fall AMC 10A | Radius r of three congruent spheres inside a right circular cone. | Geometry | 3D Geometry & Solids | 22 |
| 2021 Spring AMC 10A | Hiram's 50-page notes on 25 sheets; after removing consecutive sheets, mean of remaining pages is 19. How many sheets removed? | Algebra | Rates, Mixtures, & Averages | 22 |
| 2021 Fall AMC 10A | Number of n ≤ 50 where iterating f_1 (twice divisor count) fifty times yields 12. | Number Theory | Factors, Multiples & Divisibility | 23 |
| 2021 Spring AMC 10A | Frog on 3x3 grid with wrap-around; probability of reaching corner within 4 hops from center. | Combinatorics | Probability | 23 |
| 2021 Fall AMC 10A | Label cube edges 0 or 1; count labelings where each face sum is 2. | Combinatorics | Counting | 24 |
| 2021 Spring AMC 10A | Quadrilateral bounded by (x+ay)^2=4a^2 and (ax-y)^2=a^2; find its area in terms of a. | Geometry | Coordinate Geometry | 24 |
| 2021 Fall AMC 10A | Value of the disrespectful quadratic at x=1 that maximizes sum of roots. | Algebra | Polynomials | 25 |
| 2021 Spring AMC 10A | Place 3 red, 3 blue, 3 green chips on 3x3 grid with no two same color adjacent. How many ways? | Combinatorics | Counting | 25 |
| 2021 Fall AMC 10B | Sum of four numbers formed by cyclically permuting digits 1,2,3,4 | Algebra | Systems & Algebraic Manipulation | 1 |
| 2021 Spring AMC 10B | Count integer solutions to |x| < 3π | Algebra | Inequalities | 1 |
| 2021 Fall AMC 10B | Area of shaded quadrilateral on a 6 by 5 grid | Geometry | Coordinate Geometry | 2 |
| 2021 Spring AMC 10B | Simplify √((3-2√3)²)+√((3+2√3)²) | Algebra | Systems & Algebraic Manipulation | 2 |
| 2021 Fall AMC 10B | Compute numerator of fraction (2021/2020) - (2020/2021) in simplest form | Algebra | Systems & Algebraic Manipulation | 3 |
| 2021 Spring AMC 10B | Junior/senior debate team with percentages, find number of juniors | Algebra | Rates, Mixtures, & Averages | 3 |
| 2021 Fall AMC 10B | Temperature difference problem: Minneapolis N degrees warmer, find product of possible N after given changes | Algebra | Rates, Mixtures, & Averages | 4 |
| 2021 Spring AMC 10B | Pairs of blue/yellow shirts, find number of both-yellow pairs | Algebra | Systems & Algebraic Manipulation | 4 |
| 2021 Fall AMC 10B | Simplify n/4 where n=8^{2022} into a power of 4 | Algebra | Logarithms & Exponents | 5 |
| 2021 Spring AMC 10B | Four distinct single-digit ages multiply to 24 and 30, find sum | Number Theory | Factors, Multiples & Divisibility | 5 |
| 2021 Fall AMC 10B | Least positive integer with 2021 divisors expressed as m·6^k, find m+k | Number Theory | Factors, Multiples & Divisibility | 6 |
| 2021 Spring AMC 10B | Mean scores of two classes with ratio 3:4, find overall mean | Algebra | Rates, Mixtures, & Averages | 6 |
| 2021 Fall AMC 10B | Count distinct integers that can be sum of two special fractions a/b with a+b=15 | Combinatorics | Counting | 7 |
| 2021 Spring AMC 10B | Maximize area of region inside exactly one of four tangent circles | Geometry | Circular Geometry | 7 |
| 2021 Fall AMC 10B | Largest prime divisor of 16383; sum of its digits | Number Theory | Factors, Multiples & Divisibility | 8 |
| 2021 Spring AMC 10B | Sum of greatest and least number in second row of spiral 15×15 grid | Algebra | Sequences & Series | 8 |
| 2021 Fall AMC 10B | Fraction of red knights who are magical given ratios of colors and magical probability | Algebra | Rates, Mixtures, & Averages | 9 |
| 2021 Spring AMC 10B | Point rotated 90° around (1,5) then reflected over y=-x, find b-a | Geometry | Coordinate Geometry | 9 |
| 2021 Fall AMC 10B | Alice and Bob draw numbers; dialogue leads to deduction that Bob has 2, Alice 25, sum 27 | Algebra | Logic & Truth Values | 10 |
| 2021 Spring AMC 10B | Water poured from cone into cylinder, find new height | Geometry | 3D Geometry & Solids | 10 |
| 2021 Fall AMC 10B | Area of region bounded by six reflected arcs from sides of a regular hexagon inscribed in a circle | Geometry | Circular Geometry | 11 |
| 2021 Spring AMC 10B | Brownie pan cuts, interior pieces equal perimeter pieces, maximize total | Algebra | Systems & Algebraic Manipulation | 11 |
| 2021 Fall AMC 10B | Find condition guaranteeing x(x-y)+y(y-z)+z(z-x)=1 for integers x,y,z | Algebra | Systems & Algebraic Manipulation | 12 |
| 2021 Spring AMC 10B | Ratio of sum of odd divisors to sum of even divisors of N | Number Theory | Factors, Multiples & Divisibility | 12 |
| 2021 Fall AMC 10B | Area of isosceles triangle with two inscribed squares of side lengths 3 and 2 | Geometry | Triangles & Polygons | 13 |
| 2021 Spring AMC 10B | Base-n numeral 32d equals 263, find n+d given other numeral data | Algebra | Systems & Algebraic Manipulation | 13 |
| 2021 Fall AMC 10B | Probability that product of six dice is divisible by 4 | Combinatorics | Probability | 14 |
| 2021 Spring AMC 10B | Three parallel chords of circle lengths 38,38,34, find distance between adjacent lines | Geometry | Circular Geometry | 14 |
| 2021 Fall AMC 10B | Square with points on sides, segments intersect at right angles, find area given segment lengths 6 and 7 | Geometry | Triangles & Polygons | 15 |
| 2021 Spring AMC 10B | Given x+1/x=√5, evaluate x^11−7x^7+x^3 | Algebra | Polynomials | 15 |
| 2021 Fall AMC 10B | Expected number of balls in original positions after two random adjacent swaps on a circle of five | Combinatorics | Probability | 16 |
| 2021 Spring AMC 10B | Count uphill integers (strictly increasing digits) divisible by 15 | Combinatorics | Counting | 16 |
| 2021 Fall AMC 10B | Lines through origin, reflection of point yields another, find equation of second line | Geometry | Coordinate Geometry | 17 |
| 2021 Spring AMC 10B | Five people receive two cards from 1-10; determine true statement about cards | Algebra | Logic & Truth Values | 17 |
| 2021 Fall AMC 10B | Area of 24-sided polygon formed by three rotated squares of side 6 | Geometry | Triangles & Polygons | 18 |
| 2021 Spring AMC 10B | Probability every even number appears before first odd when rolling a fair die | Combinatorics | Probability | 18 |
| 2021 Fall AMC 10B | Sum of leading digits of r-th root of a 313-digit repunit of 7s for r=2..6 | Algebra | Logarithms & Exponents | 19 |
| 2021 Spring AMC 10B | Find average of all integers in S given average after removing max/min | Algebra | Rates, Mixtures, & Averages | 19 |
| 2021 Fall AMC 10B | Conditional probability that Hugo's first roll was 5 given he won the dice game | Combinatorics | Probability | 20 |
| 2021 Spring AMC 10B | Area of pentagon built from 11 segments of length 2, express as √m+√n | Geometry | Triangles & Polygons | 20 |
| 2021 Fall AMC 10B | Number of interior intersection points of sides of four regular polygons (5,6,7,8) inscribed in same circle | Combinatorics | Counting | 21 |
| 2021 Spring AMC 10B | Folded square paper, find perimeter of triangle formed after fold | Geometry | Triangles & Polygons | 21 |
| 2021 Fall AMC 10B | Sum of ten smallest n where sum of products jk for 1≤j<k≤n is divisible by 3 | Number Theory | Modular Arithmetic | 22 |
| 2021 Spring AMC 10B | Probability at least one box gets three blocks same color among 5 colors | Combinatorics | Probability | 22 |
| 2021 Fall AMC 10B | Probability that a monochromatic triangle exists when coloring sides and diagonals of a regular pentagon red/blue | Combinatorics | Probability | 23 |
| 2021 Spring AMC 10B | Probability coin covers black region on square with corner triangles and diamond | Geometry | Coordinate Geometry | 23 |
| 2021 Fall AMC 10B | Number of distinct 2x2x2 cubes with 4 white and 4 blue unit cubes up to rotation | Combinatorics | Counting | 24 |
| 2021 Spring AMC 10B | Find brick-wall configuration where second player has a winning strategy | Combinatorics | Counting | 24 |
| 2021 Fall AMC 10B | Sum of all possible areas of rectangle R inscribed in larger square with given smaller shapes | Geometry | Triangles & Polygons | 25 |
| 2021 Spring AMC 10B | Lattice points 1≤x,y≤30 below line y=mx; length of m-interval for 300 points | Geometry | Coordinate Geometry | 25 |
| 2020 AMC 10A | Solve linear equation x - 3/4 = 5/12 - 1/3 | Algebra | Systems & Algebraic Manipulation | 1 |
| 2020 AMC 10A | Given average of five numbers, find average of two of them | Algebra | Rates, Mixtures, & Averages | 2 |
| 2020 AMC 10A | Simplify product of three fractions involving a, b, c | Algebra | Systems & Algebraic Manipulation | 3 |
| 2020 AMC 10A | Find net rate of pay per hour given distance, fuel efficiency, and pay rates | Algebra | Rates, Mixtures, & Averages | 4 |
| 2020 AMC 10A | Sum of all real solutions to |x^2 - 12x + 34| = 2 | Algebra | Systems & Algebraic Manipulation | 5 |
| 2020 AMC 10A | Count 4-digit numbers with only even digits that are divisible by 5 | Combinatorics | Counting | 6 |
| 2020 AMC 10A | Find common row sum of a 5x5 magic square using numbers -10 to 14 | Algebra | Sequences & Series | 7 |
| 2020 AMC 10A | Evaluate alternating sum 1+2+3-4+...+197+198+199-200 | Algebra | Sequences & Series | 8 |
| 2020 AMC 10A | Least number of bench sections for equal adults and children given capacities | Number Theory | Factors, Multiples & Divisibility | 9 |
| 2020 AMC 10A | Total surface area of a tower of seven stacked cubes with volumes 1^3 to 7^3 | Geometry | 3D Geometry & Solids | 10 |
| 2020 AMC 10A | Median of concatenated list 1,...,2020 and 1^2,...,2020^2 | Algebra | Sequences & Series | 11 |
| 2020 AMC 10A | Area of isosceles triangle with perpendicular medians of length 12 | Geometry | Triangles & Polygons | 12 |
| 2020 AMC 10A | Probability a random-walking frog hits a vertical side of a 4x4 square | Combinatorics | Probability | 13 |
| 2020 AMC 10A | Evaluate algebraic expression given x+y=4 and xy=-2 | Algebra | Systems & Algebraic Manipulation | 14 |
| 2020 AMC 10A | Probability a randomly chosen divisor of 12! is a perfect square | Number Theory | Factors, Multiples & Divisibility | 15 |
| 2020 AMC 10A | Find distance d such that half of a square is within d of a lattice point | Geometry | Coordinate Geometry | 16 |
| 2020 AMC 10A | Number of integers n with P(n)<=0 where P(x)=∏(x-k^2) | Algebra | Polynomials | 17 |
| 2020 AMC 10A | Count ordered quadruples from {0,1,2,3} making ad-bc odd | Combinatorics | Counting | 18 |
| 2020 AMC 10A | Number of face paths from top to bottom of a regular dodecahedron | Combinatorics | Counting | 19 |
| 2020 AMC 10A | Area of quadrilateral with two right angles and given side lengths | Geometry | Triangles & Polygons | 20 |
| 2020 AMC 10A | Express (2^289+1)/(2^17+1) as sum of distinct powers of two | Algebra | Polynomials | 21 |
| 2020 AMC 10A | Count n≤1000 such that sum of floors of 998/n, 999/n, 1000/n not divisible by 3 | Number Theory | Modular Arithmetic | 22 |
| 2020 AMC 10A | Sequences of three isometries that return a given triangle to its position | Combinatorics | Counting | 23 |
| 2020 AMC 10A | Find smallest n>1000 with gcd conditions; sum its digits | Number Theory | Factors, Multiples & Divisibility | 24 |
| 2020 AMC 10A | Probability optimal reroll strategy uses exactly two dice in a die game | Combinatorics | Probability | 25 |
| 2020 AMC 10B | Evaluate 1-(-2)-3-(-4)-5-(-6) | Algebra | Systems & Algebraic Manipulation | 1 |
| 2020 AMC 10B | Total volume of 5 cubes side 1 and 5 cubes side 2 | Geometry | 3D Geometry & Solids | 2 |
| 2020 AMC 10B | Given w:x=4:3, y:z=3:2, z:x=1:6, find w:y | Algebra | Systems & Algebraic Manipulation | 3 |
| 2020 AMC 10B | Right triangle with prime acute angles, find least possible smaller prime | Number Theory | Factors, Multiples & Divisibility | 4 |
| 2020 AMC 10B | Arrange 1 brown, 1 purple, 2 green, 3 yellow tiles in a row | Combinatorics | Counting | 5 |
| 2020 AMC 10B | Odometer palindrome 15951, find average speed to next palindrome in 2 hours | Algebra | Rates, Mixtures, & Averages | 6 |
| 2020 AMC 10B | Count even multiples of 3 less than 2020 that are perfect squares | Number Theory | Factors, Multiples & Divisibility | 7 |
| 2020 AMC 10B | Locations for R such that triangle PQR right with area 12, PQ=8 | Geometry | Triangles & Polygons | 8 |
| 2020 AMC 10B | Integer pairs (x,y) satisfying x^2020 + y^2 = 2y | Algebra | Polynomials | 9 |
| 2020 AMC 10B | Three-quarter sector of circle radius 4 rolled into cone, find volume | Geometry | 3D Geometry & Solids | 10 |
| 2020 AMC 10B | Probability two people pick exactly 2 same books out of 10, each picking 5 | Combinatorics | Probability | 11 |
| 2020 AMC 10B | Number of leading zeros in decimal expansion of 1/20^20 | Algebra | Logarithms & Exponents | 12 |
| 2020 AMC 10B | Andy ant moves 1,2,3,... turning left, location at 2020th turn | Algebra | Sequences & Series | 13 |
| 2020 AMC 10B | Shaded area inside hexagon outside six semicircles on sides | Geometry | Circular Geometry | 14 |
| 2020 AMC 10B | Erasing every 3rd,4th,5th digit from repeated 12345, sum at positions 2019-2021 | Combinatorics | Counting | 15 |
| 2020 AMC 10B | Bela and Jenn choose numbers >1 apart on [0,n], who wins? | Combinatorics | Counting | 16 |
| 2020 AMC 10B | Pair 10 people in circle into 5 mutually-knowing pairs | Combinatorics | Counting | 17 |
| 2020 AMC 10B | Urn with 1 red,1 blue; add same color ball 4 times, probability 3 of each | Combinatorics | Probability | 18 |
| 2020 AMC 10B | Find missing digit A in number 158A00A4AA0 equal to C(52,10) | Number Theory | Modular Arithmetic | 19 |
| 2020 AMC 10B | Volume of points within distance r of a 1x3x4 prism, find bc/ad | Geometry | 3D Geometry & Solids | 20 |
| 2020 AMC 10B | Square with points E,F,G,H,I,J, all regions area 1, find FI^2 | Geometry | Triangles & Polygons | 21 |
| 2020 AMC 10B | Remainder of 2^202+202 divided by 2^101+2^51+1 | Algebra | Polynomials | 22 |
| 2020 AMC 10B | Count sequences of 20 transformations L,R,H,V returning square to original | Combinatorics | Counting | 23 |
| 2020 AMC 10B | Number of positive integers n with (n+1000)/70 = floor(sqrt(n)) | Algebra | Inequalities | 24 |
| 2020 AMC 10B | Number of ordered factorizations of 96 into factors >1 | Combinatorics | Counting | 25 |
| 2019 AMC 10A | Evaluate 2^(0^(1^9)) + ((2^0)^1)^9. | Algebra | Logarithms & Exponents | 1 |
| 2019 AMC 10A | Find the hundreds digit of 20! − 15!. | Number Theory | Modular Arithmetic | 2 |
| 2019 AMC 10A | Given ages last year and this year, find the age difference of Ana and Bonita. | Algebra | Rates, Mixtures, & Averages | 3 |
| 2019 AMC 10A | Minimum balls drawn to guarantee 15 of one color from a box of mixed colors. | Combinatorics | Counting | 4 |
| 2019 AMC 10A | Greatest number of consecutive integers whose sum equals 45. | Algebra | Sequences & Series | 5 |
| 2019 AMC 10A | How many listed quadrilateral types have a point equidistant from all four vertices. | Geometry | Circular Geometry | 6 |
| 2019 AMC 10A | Area of triangle enclosed by two lines through (2,2) and the line x+y=10. | Geometry | Coordinate Geometry | 7 |
| 2019 AMC 10A | Number of rigid motions that map an infinite square-and-segment pattern onto itself. | Geometry | Triangles & Polygons | 8 |
| 2019 AMC 10A | Greatest three-digit n such that sum of first n integers does not divide n!. | Number Theory | Factors, Multiples & Divisibility | 9 |
| 2019 AMC 10A | Number of 1-foot tiles visited by a diagonal bug on a 10 by 17 foot floor. | Number Theory | Factors, Multiples & Divisibility | 10 |
| 2019 AMC 10A | Number of divisors of 201^9 that are perfect squares or perfect cubes. | Number Theory | Factors, Multiples & Divisibility | 11 |
| 2019 AMC 10A | Compare the mean, median, and median of the modes of 2019 calendar dates. | Algebra | Rates, Mixtures, & Averages | 12 |
| 2019 AMC 10A | In isosceles triangle ABC with circle on BC, find angle BFC formed by diagonals. | Geometry | Circular Geometry | 13 |
| 2019 AMC 10A | Sum of all possible numbers of distinct intersection points of four lines. | Combinatorics | Counting | 14 |
| 2019 AMC 10A | Find p+q for a recursively defined sequence term a_{2019}=p/q. | Algebra | Sequences & Series | 15 |
| 2019 AMC 10A | Area inside a large circle that surrounds 13 tangent unit circles. | Geometry | Circular Geometry | 16 |
| 2019 AMC 10A | Number of distinct towers of height 8 using 2 red, 3 blue, 4 green cubes. | Combinatorics | Counting | 17 |
| 2019 AMC 10A | Find base k such that 0.232323..._k equals 7/51. | Algebra | Sequences & Series | 18 |
| 2019 AMC 10A | Minimum value of (x+1)(x+2)(x+3)(x+4)+2019 for real x. | Algebra | Inequalities | 19 |
| 2019 AMC 10A | Probability that each row and column sum is odd when numbers 1-9 fill a 3x3 grid. | Combinatorics | Probability | 20 |
| 2019 AMC 10A | Distance from a sphere's center to a triangle plane when triangle sides are tangent to sphere. | Geometry | 3D Geometry & Solids | 21 |
| 2019 AMC 10A | Probability |x-y| > 1/2 when numbers are chosen by coin flips or uniformly. | Combinatorics | Probability | 22 |
| 2019 AMC 10A | 2019th number spoken by Tadd in a rotating counting game with increasing counts. | Algebra | Sequences & Series | 23 |
| 2019 AMC 10A | Evaluate 1/A + 1/B + 1/C from partial fraction decomposition of cubic reciprocal. | Algebra | Polynomials | 24 |
| 2019 AMC 10A | How many integers n in 1..50 make (n^2−1)!/(n!)^n an integer. | Number Theory | Factors, Multiples & Divisibility | 25 |
| 2019 AMC 10B | Alicia pours water between two containers; find ratio of volumes. | Algebra | Rates, Mixtures, & Averages | 1 |
| 2019 AMC 10B | Find counterexample to 'if n not prime then n-2 is prime'. | Number Theory | Factors, Multiples & Divisibility | 2 |
| 2019 AMC 10B | Find number of non-seniors playing instrument given percentages. | Algebra | Rates, Mixtures, & Averages | 3 |
| 2019 AMC 10B | Find common point of lines with coefficients in arithmetic progression. | Algebra | Systems & Algebraic Manipulation | 4 |
| 2019 AMC 10B | Determine which statement about reflected triangle is not always true. | Geometry | Coordinate Geometry | 5 |
| 2019 AMC 10B | Solve factorial equation for n and find digit sum. | Algebra | Polynomials | 6 |
| 2019 AMC 10B | Find minimum n such that n purple candies cost equals cost of 12,14,15 other candies. | Number Theory | Factors, Multiples & Divisibility | 7 |
| 2019 AMC 10B | Find shaded area inside square with four equilateral triangles. | Geometry | Triangles & Polygons | 8 |
| 2019 AMC 10B | Find range of function with floor and absolute value. | Algebra | Functions | 9 |
| 2019 AMC 10B | How many points C make triangle with given perimeter and area? | Geometry | Triangles & Polygons | 10 |
| 2019 AMC 10B | Find difference in blue marbles between two jars given ratios and totals. | Algebra | Rates, Mixtures, & Averages | 11 |
| 2019 AMC 10B | Find maximum digit sum in base-7 representation of number less than 2019. | Number Theory | Modular Arithmetic | 12 |
| 2019 AMC 10B | Find x such that median equals mean of set {4,6,8,17,x}. | Algebra | Rates, Mixtures, & Averages | 13 |
| 2019 AMC 10B | Find sum of missing digits in 19! using divisibility rules. | Number Theory | Modular Arithmetic | 14 |
| 2019 AMC 10B | Two right triangles sharing two sides; find square of product of third sides. | Geometry | Triangles & Polygons | 15 |
| 2019 AMC 10B | Find ratio AD:DB in right triangle with given equalities. | Geometry | Triangles & Polygons | 16 |
| 2019 AMC 10B | Probability red ball in higher bin than green ball given geometric distribution. | Combinatorics | Probability | 17 |
| 2019 AMC 10B | Henry walks 3/4 distance repeatedly; find difference between limiting points. | Algebra | Sequences & Series | 18 |
| 2019 AMC 10B | Count distinct products of two distinct divisors of 100,000. | Number Theory | Factors, Multiples & Divisibility | 19 |
| 2019 AMC 10B | Find shaded area inside circle overlapping three semicircles. | Geometry | Circular Geometry | 20 |
| 2019 AMC 10B | Probability of getting two heads in a row with second tail seen before second head. | Combinatorics | Probability | 21 |
| 2019 AMC 10B | Probability three players each have $1 after 2019 rounds of random giving. | Combinatorics | Probability | 22 |
| 2019 AMC 10B | Find area of circle given two points and tangents intersecting on x-axis. | Geometry | Circular Geometry | 23 |
| 2019 AMC 10B | Find interval containing least m with x_m ≤ 4+1/2^20 for recursive sequence. | Algebra | Sequences & Series | 24 |
| 2019 AMC 10B | Count binary sequences length 19 with given restrictions. | Combinatorics | Counting | 25 |
| 2018 AMC 10A | Evaluate a nested expression with negative exponents and fractions. | Algebra | Systems & Algebraic Manipulation | 1 |
| 2018 AMC 10A | Compare soda amounts given percent relationships among three people. | Algebra | Rates, Mixtures, & Averages | 2 |
| 2018 AMC 10A | Find the date when 10! seconds after noon Jan 1 elapses. | Algebra | Rates, Mixtures, & Averages | 3 |
| 2018 AMC 10A | Schedule 3 distinct math courses in 6 periods with no two consecutive. | Combinatorics | Counting | 4 |
| 2018 AMC 10A | Determine possible distance d given three false statements about the distance. | Algebra | Logic & Truth Values | 5 |
| 2018 AMC 10A | Find total votes given 65% likes and a score of 90. | Algebra | Rates, Mixtures, & Averages | 6 |
| 2018 AMC 10A | Count integer n making 4000*(2/5)^n an integer. | Number Theory | Factors, Multiples & Divisibility | 7 |
| 2018 AMC 10A | Solve coin word problem with 23 coins, total 320 cents, find difference. | Algebra | Systems & Algebraic Manipulation | 8 |
| 2018 AMC 10A | Find area of trapezoid in diagram of similar isosceles triangles. | Geometry | Triangles & Polygons | 9 |
| 2018 AMC 10A | Given sqrt(49-x^2)-sqrt(25-x^2)=3, find sqrt(49-x^2)+sqrt(25-x^2). | Algebra | Systems & Algebraic Manipulation | 10 |
| 2018 AMC 10A | Find numerator n of probability that sum of 7 dice is 10. | Combinatorics | Counting | 11 |
| 2018 AMC 10A | Count solutions to system with absolute values: x+3y=3, ||x|-|y||=1. | Algebra | Systems & Algebraic Manipulation | 12 |
| 2018 AMC 10A | Find crease length when folding a 3-4-5 triangle so A meets B. | Geometry | Triangles & Polygons | 13 |
| 2018 AMC 10A | Find floor of (3^100+2^100)/(3^96+2^96). | Algebra | Logarithms & Exponents | 14 |
| 2018 AMC 10A | Find distance AB between tangency points of two circles inside a larger circle. | Geometry | Circular Geometry | 15 |
| 2018 AMC 10A | Count integer-length segments from right-angle vertex to hypotenuse. | Geometry | Triangles & Polygons | 16 |
| 2018 AMC 10A | Find least possible element of a 6-element subset of {1..12} with no multiples. | Combinatorics | Counting | 17 |
| 2018 AMC 10A | Count nonnegative integers expressible as sum a_i*3^i with a_i in {-1,0,1}. | Combinatorics | Counting | 18 |
| 2018 AMC 10A | Find probability m^n ends in 1; m from {11,13,15,17,19}, n from 1999-2018. | Combinatorics | Probability | 19 |
| 2018 AMC 10A | Count symmetric 7x7 scanning codes with at least one square of each color. | Combinatorics | Counting | 20 |
| 2018 AMC 10A | Find a for circle x^2+y^2=a^2 and parabola y=x^2-a intersecting at exactly 3 points. | Geometry | Coordinate Geometry | 21 |
| 2018 AMC 10A | Given GCD constraints among a,b,c,d, identify divisor of a. | Number Theory | Factors, Multiples & Divisibility | 22 |
| 2018 AMC 10A | Find fraction of right triangular field planted, leaving a square corner unplanted. | Geometry | Triangles & Polygons | 23 |
| 2018 AMC 10A | Find area of quadrilateral FDBG in triangle with midpoints and angle bisector. | Geometry | Triangles & Polygons | 24 |
| 2018 AMC 10A | Maximize a+b+c for C_n-B_n=A_n^2 with repeated-digit numbers, at least two n. | Number Theory | Modular Arithmetic | 25 |
| 2018 AMC 10B | How many 2-inch squares from a 20x18 inch cornbread? | Algebra | Systems & Algebraic Manipulation | 1 |
| 2018 AMC 10B | Average speed in last 30 min of a 90-min trip given first two segments. | Algebra | Rates, Mixtures, & Averages | 2 |
| 2018 AMC 10B | Different values of (a×b)+(c×d) using digits 1-4 once. | Combinatorics | Counting | 3 |
| 2018 AMC 10B | Find X+Y+Z for box with face areas 24,24,48,48,72,72. | Algebra | Systems & Algebraic Manipulation | 4 |
| 2018 AMC 10B | Subsets of {2,...,9} containing at least one prime. | Combinatorics | Counting | 5 |
| 2018 AMC 10B | Probability 3 draws needed until sum of chips exceeds 4. | Combinatorics | Probability | 6 |
| 2018 AMC 10B | Number of small semicircles on diameter given area ratio 1:18. | Geometry | Circular Geometry | 7 |
| 2018 AMC 10B | Steps in toothpick staircase using 180 toothpicks. | Algebra | Sequences & Series | 8 |
| 2018 AMC 10B | Sum with same probability as 10 when rolling seven dice. | Combinatorics | Probability | 9 |
| 2018 AMC 10B | Volume of pyramid with apex at midpoint of edge in rectangular box. | Geometry | 3D Geometry & Solids | 10 |
| 2018 AMC 10B | Which expression p^2+... is never prime for prime p? | Number Theory | Modular Arithmetic | 11 |
| 2018 AMC 10B | Area traced by centroid of triangle on circle diameter AB=24. | Geometry | Circular Geometry | 12 |
| 2018 AMC 10B | Count of numbers 10^n+1 divisible by 101 in first 2018 terms. | Number Theory | Modular Arithmetic | 13 |
| 2018 AMC 10B | Minimum distinct values in 2018 numbers with unique mode appearing 10 times. | Algebra | Rates, Mixtures, & Averages | 14 |
| 2018 AMC 10B | Area of square wrapping paper folded over a box with base w and height h. | Geometry | Triangles & Polygons | 15 |
| 2018 AMC 10B | Sum of cubes mod 6 of strictly increasing sequence summing to 2018^{2018}. | Number Theory | Modular Arithmetic | 16 |
| 2018 AMC 10B | Side length of equilateral octagon inscribed in rectangle of width 8, height 6. | Geometry | Triangles & Polygons | 17 |
| 2018 AMC 10B | Seating arrangements of 3 sibling pairs in a van with restrictions. | Combinatorics | Counting | 18 |
| 2018 AMC 10B | Sum of digits of Joey's age when his age next multiple of Zoe's. | Number Theory | Factors, Multiples & Divisibility | 19 |
| 2018 AMC 10B | Find f(2018) for recursive sequence f(n)=f(n-1)-f(n-2)+n. | Algebra | Sequences & Series | 20 |
| 2018 AMC 10B | Smallest next divisor after 323 of an even 4-digit number. | Number Theory | Factors, Multiples & Divisibility | 21 |
| 2018 AMC 10B | Probability that x, y, 1 form an obtuse triangle for x,y in [0,1]. | Geometry | Triangles & Polygons | 22 |
| 2018 AMC 10B | Ordered pairs (a,b) satisfying equation with lcm and gcd. | Number Theory | Factors, Multiples & Divisibility | 23 |
| 2018 AMC 10B | Area of intersection of two triangles in a regular hexagon of side 1. | Geometry | Triangles & Polygons | 24 |
| 2018 AMC 10B | Number of real solutions to x^2+10000⌊x⌋=10000x. | Algebra | Systems & Algebraic Manipulation | 25 |
| 2017 AMC 10A | Value of (2(2(2(2(2(2+1)+1)+1)+1)+1)+1) | Algebra | Sequences & Series | 1 |
| 2017 AMC 10A | Maximize popsicles bought with $8 given single $1, 3-pack $2, 5-pack $3 | Algebra | Rates, Mixtures, & Averages | 2 |
| 2017 AMC 10A | Total walkway area around six 6x2 ft flower beds in a 3x2 grid with 1-ft walkways | Geometry | Triangles & Polygons | 3 |
| 2017 AMC 10A | Mom puts 3 toys in box per 30 sec, child removes 2; time to put all 30 toys in box | Algebra | Rates, Mixtures, & Averages | 4 |
| 2017 AMC 10A | Sum of two numbers equals 4 times their product; find sum of reciprocals | Algebra | Systems & Algebraic Manipulation | 5 |
| 2017 AMC 10A | If all multiple choice right then get A; which statement logically follows? | Algebra | Logic & Truth Values | 6 |
| 2017 AMC 10A | Jerry walks two sides of square, Silvia diagonal; percent shorter is Silvia's trip? | Geometry | Triangles & Polygons | 7 |
| 2017 AMC 10A | 30 people: 20 know each other, 10 know no one; hugs for known, handshakes for unknown; count handshakes | Combinatorics | Counting | 8 |
| 2017 AMC 10A | Minnie and Penny travel same route opposite ways with different speeds; time difference | Algebra | Rates, Mixtures, & Averages | 9 |
| 2017 AMC 10A | Choose fourth rod from 1-30 to form quadrilateral with 3,7,15; how many choices? | Geometry | Triangles & Polygons | 10 |
| 2017 AMC 10A | Find length of segment AB given volume of region within 3 units of AB is 216π | Geometry | 3D Geometry & Solids | 11 |
| 2017 AMC 10A | Set S of (x,y) such that two of {3, x+2, y-4} equal and third ≤; describe S | Geometry | Coordinate Geometry | 12 |
| 2017 AMC 10A | Sequence F_n mod 3; sum of 8 consecutive terms starting F_2017 | Algebra | Sequences & Series | 13 |
| 2017 AMC 10A | Ticket costs 20% of (allowance - soda), soda 5% of (allowance - ticket); fraction spent | Algebra | Systems & Algebraic Manipulation | 14 |
| 2017 AMC 10A | Chloe picks x in [0,2017], Laurent y in [0,4034]; probability y > x | Combinatorics | Probability | 15 |
| 2017 AMC 10A | Find smallest T where at least 5 of 10 horses (lap times 1..10) meet at start | Number Theory | Factors, Multiples & Divisibility | 16 |
| 2017 AMC 10A | Points P,Q,R,S on circle x^2+y^2=25 with integer coords; maximize irrational distance ratio PQ/RS | Geometry | Coordinate Geometry | 17 |
| 2017 AMC 10A | Amelia and Blaine alternate coin tosses, different head probabilities; Amelia wins probability | Combinatorics | Probability | 18 |
| 2017 AMC 10A | Seat 5 people in a row: Alice not next to Bob/Carla, Derek not next to Eric | Combinatorics | Counting | 19 |
| 2017 AMC 10A | S(n) is digit sum; S(n)=1274; which could be S(n+1)? | Number Theory | Modular Arithmetic | 20 |
| 2017 AMC 10A | Two squares inscribed in 3-4-5 right triangles: one at right angle, one on hypotenuse; ratio of sides | Geometry | Triangles & Polygons | 21 |
| 2017 AMC 10A | In equilateral triangle, sides AB, AC tangent to circle at B,C; fraction of area outside circle | Geometry | Circular Geometry | 22 |
| 2017 AMC 10A | How many triangles with vertices at integer coordinates (i,j) for 1≤i,j≤5? | Combinatorics | Counting | 23 |
| 2017 AMC 10A | Polynomial g has 3 distinct roots, each root of f; find f(1) | Algebra | Polynomials | 24 |
| 2017 AMC 10A | How many 3-digit numbers have a permutation that is a multiple of 11? | Combinatorics | Counting | 25 |
| 2017 AMC 10B | Mary multiplies two-digit number by 3, adds 11, reverses digits; find original number. | Algebra | Systems & Algebraic Manipulation | 1 |
| 2017 AMC 10B | Sofia runs 5 laps with varying speeds; find total time. | Algebra | Rates, Mixtures, & Averages | 2 |
| 2017 AMC 10B | Given inequalities for real numbers x, y, z; determine which expression is always positive. | Algebra | Inequalities | 3 |
| 2017 AMC 10B | Given fractional equation, find value of another fractional expression. | Algebra | Systems & Algebraic Manipulation | 4 |
| 2017 AMC 10B | Jellybean count problem: find original number of blueberry given ratios after eating. | Algebra | Systems & Algebraic Manipulation | 5 |
| 2017 AMC 10B | Maximum number of solid 2x2x1 blocks that fit in a 3x2x3 box. | Geometry | 3D Geometry & Solids | 6 |
| 2017 AMC 10B | Samia bikes and walks equal distances; find distance walked given total time. | Algebra | Rates, Mixtures, & Averages | 7 |
| 2017 AMC 10B | Find coordinates of point C given triangle vertices A, B, altitude foot D, and AB=AC. | Geometry | Coordinate Geometry | 8 |
| 2017 AMC 10B | Contestant guesses on 3 multiple-choice questions; probability of at least 2 correct. | Combinatorics | Probability | 9 |
| 2017 AMC 10B | Two perpendicular lines intersect at (1,-5); find constant c. | Algebra | Systems & Algebraic Manipulation | 10 |
| 2017 AMC 10B | Survey: find fraction of students who say they dislike dancing but actually like it. | Algebra | Rates, Mixtures, & Averages | 11 |
| 2017 AMC 10B | Car fuel efficiency and price comparison; find percent savings. | Algebra | Rates, Mixtures, & Averages | 12 |
| 2017 AMC 10B | Students taking classes; given numbers in each and overlap, find number taking all three. | Combinatorics | Counting | 13 |
| 2017 AMC 10B | Random N from 1 to 2020; find probability N^16 mod 5 is 1. | Number Theory | Modular Arithmetic | 14 |
| 2017 AMC 10B | Rectangle ABCD, E foot from B to AC; find area of triangle AED. | Geometry | Triangles & Polygons | 15 |
| 2017 AMC 10B | Count positive integers ≤2017 containing digit 0. | Combinatorics | Counting | 16 |
| 2017 AMC 10B | Count monotonous numbers (digits strictly increasing or decreasing). | Combinatorics | Counting | 17 |
| 2017 AMC 10B | Paint 6 disks 3 blue, 2 red, 1 green; rotations/reflections considered same; count paintings. | Combinatorics | Counting | 18 |
| 2017 AMC 10B | Equilateral triangle extended; find ratio of areas of new triangle to original. | Geometry | Triangles & Polygons | 19 |
| 2017 AMC 10B | Random divisor of 21! chosen; probability it is odd. | Number Theory | Factors, Multiples & Divisibility | 20 |
| 2017 AMC 10B | Right triangle with D midpoint of hypotenuse; find sum of inradii of two smaller triangles. | Geometry | Triangles & Polygons | 21 |
| 2017 AMC 10B | Circle diameter extended, perpendicular drawn; find area of triangle formed. | Geometry | Circular Geometry | 22 |
| 2017 AMC 10B | N is integer 123...44; find remainder upon division by 45. | Number Theory | Modular Arithmetic | 23 |
| 2017 AMC 10B | Equilateral triangle with vertices on xy=1 and centroid at hyperbola vertex; find squared area. | Geometry | Coordinate Geometry | 24 |
| 2017 AMC 10B | Isabella's 7 test scores distinct 91-100, averages integers; 7th=95; find 6th. | Number Theory | Modular Arithmetic | 25 |
| 2016 AMC 10A | Evaluate the expression (11! - 10!) / 9! | Algebra | Systems & Algebraic Manipulation | 1 |
| 2016 AMC 10A | Find x such that 10^x * 100^(2x) = 1000^5 | Algebra | Logarithms & Exponents | 2 |
| 2016 AMC 10A | Ben spent $12.50 more than David on bagels, with David spending 25¢ less per dollar; find total spent | Algebra | Rates, Mixtures, & Averages | 3 |
| 2016 AMC 10A | Evaluate the remainder function rem(3/8, -2/5) using the given definition with floor | Algebra | Functions | 4 |
| 2016 AMC 10A | Rectangular box with side ratio 1:3:4 has integer volume; which given volume is possible? | Geometry | 3D Geometry & Solids | 5 |
| 2016 AMC 10A | Ximena lists 1..30, Emilio replaces each '2' digit with '1'; find difference in sums | Combinatorics | Counting | 6 |
| 2016 AMC 10A | Given 7 data values where mean, median, and mode all equal x, find x | Algebra | Rates, Mixtures, & Averages | 7 |
| 2016 AMC 10A | Trickster Rabbit doubles Fox's money each crossing minus 40 coins; after 3 crossings money is gone; find starting amount | Algebra | Systems & Algebraic Manipulation | 8 |
| 2016 AMC 10A | Triangular array of 2016 coins, N rows; find sum of digits of N | Algebra | Sequences & Series | 9 |
| 2016 AMC 10A | Rug with three concentric rectangles, areas in arithmetic progression; find inner rectangle length | Algebra | Sequences & Series | 10 |
| 2016 AMC 10A | Find the area of the shaded L-shaped region in a 8×5 rectangle | Geometry | Triangles & Polygons | 11 |
| 2016 AMC 10A | Probability that product of three distinct integers from 1 to 2016 is odd | Combinatorics | Probability | 12 |
| 2016 AMC 10A | Five friends in a row of seats move; find Ada's original seat given constraints | Algebra | Logic & Truth Values | 13 |
| 2016 AMC 10A | Number of ways to write 2016 as a sum of twos and threes, ignoring order | Combinatorics | Counting | 14 |
| 2016 AMC 10A | Seven radius-1 cookies cut from a larger circle; find the radius of the scrap cookie | Geometry | Circular Geometry | 15 |
| 2016 AMC 10A | Triangle reflected across x-axis then rotated 90° CCW; which transformation returns it to original? | Geometry | Coordinate Geometry | 16 |
| 2016 AMC 10A | Probability that at least 3/5 of N green balls are on same side of a red ball; find smallest N with P<321/400 | Combinatorics | Probability | 17 |
| 2016 AMC 10A | Assign numbers 1..8 to cube vertices so each face sums the same; count distinct arrangements | Combinatorics | Counting | 18 |
| 2016 AMC 10A | Rectangle ABCD with points E,F on BC; find ratio BP:PQ:QD where AE,AF intersect BD | Geometry | Triangles & Polygons | 19 |
| 2016 AMC 10A | Expansion of (a+b+c+d+1)^N has exactly 1001 terms with all four variables; find N | Combinatorics | Counting | 20 |
| 2016 AMC 10A | Three circles tangent to a line and each other; find area of triangle formed by their centers | Geometry | Circular Geometry | 21 |
| 2016 AMC 10A | 110n³ has exactly 110 divisors; how many divisors does 81n⁴ have? | Number Theory | Factors, Multiples & Divisibility | 22 |
| 2016 AMC 10A | Binary operation ◊ with given properties; solve 2016 ◊ (6 ◊ x) = 100 for x | Algebra | Functions | 23 |
| 2016 AMC 10A | Quadrilateral inscribed in circle of radius 200√2 with three sides 200; find the fourth side | Geometry | Circular Geometry | 24 |
| 2016 AMC 10A | Ordered triples of positive integers with given pairwise LCMs: lcm(x,y)=72, lcm(x,z)=600, lcm(y,z)=900 | Number Theory | Factors, Multiples & Divisibility | 25 |
| 2016 AMC 10B | Evaluate (2a^{-1}+a^{-1}/2)/a for a=1/2 | Algebra | Systems & Algebraic Manipulation | 1 |
| 2016 AMC 10B | Evaluate (2♡4)/(4♡2) for custom operation n♡m=n^3 m^2 | Algebra | Functions | 2 |
| 2016 AMC 10B | Evaluate nested absolute value expression with x=-2016 | Algebra | Systems & Algebraic Manipulation | 3 |
| 2016 AMC 10B | Zoey reads 15 books, each takes 1 more day, finished 1st on Mon, 2nd Wed, find day of 15th | Number Theory | Modular Arithmetic | 4 |
| 2016 AMC 10B | Given mean and median of 4 cousins' ages, find sum of youngest and oldest | Algebra | Rates, Mixtures, & Averages | 5 |
| 2016 AMC 10B | Two three-digit numbers with all six digits distinct, sum is three-digit, minimize digit sum of sum | Algebra | Systems & Algebraic Manipulation | 6 |
| 2016 AMC 10B | Two acute angles ratio 5:4, complement of one twice complement of other, find sum | Algebra | Systems & Algebraic Manipulation | 7 |
| 2016 AMC 10B | Find tens digit of 2015^{2016}-2017 | Number Theory | Modular Arithmetic | 8 |
| 2016 AMC 10B | Triangle vertices on y=x^2, A at origin, base parallel to x-axis, area 64, find base length | Geometry | Coordinate Geometry | 9 |
| 2016 AMC 10B | Weight of equilateral triangle side 3 is 12 oz, find weight of similar triangle side 5 (same thickness) | Geometry | Triangles & Polygons | 10 |
| 2016 AMC 10B | 20 fence posts, corners, rest evenly spaced 4 yards, longer side has twice as many posts as shorter, find area | Algebra | Systems & Algebraic Manipulation | 11 |
| 2016 AMC 10B | Two different numbers from {1,2,3,4,5} multiplied, probability product even | Combinatorics | Probability | 12 |
| 2016 AMC 10B | 1000 babies from twins, triplets, quadruplets; ratios given, find how many were in quadruplet sets | Algebra | Systems & Algebraic Manipulation | 13 |
| 2016 AMC 10B | Number of axis-aligned integer-coordinate squares inside region bounded by y=πx, y=-0.1, x=5.1 | Geometry | Coordinate Geometry | 14 |
| 2016 AMC 10B | Numbers 1-9 in 3x3 grid, consecutive numbers share an edge, corners sum 18, find center number | Algebra | Logic & Truth Values | 15 |
| 2016 AMC 10B | Infinite geometric series with second term 1, find minimum possible sum S | Algebra | Sequences & Series | 16 |
| 2016 AMC 10B | Numbers 2-7 on cube faces, each vertex product of three adjoining faces, maximize sum of eight products | Algebra | Systems & Algebraic Manipulation | 17 |
| 2016 AMC 10B | Number of ways to write 345 as sum of two or more consecutive positive integers | Number Theory | Factors, Multiples & Divisibility | 18 |
| 2016 AMC 10B | Rectangle ABCD with points E,F,G, segments AG,AC intersect EF at Q,P; find PQ/EF | Geometry | Triangles & Polygons | 19 |
| 2016 AMC 10B | Dilation sends circle radius 2 center (2,2) to radius 3 center (5,6); find distance origin moves | Geometry | Coordinate Geometry | 20 |
| 2016 AMC 10B | Area enclosed by graph of x^2+y^2=|x|+|y| | Geometry | Coordinate Geometry | 21 |
| 2016 AMC 10B | 21 teams, each won 10 lost 10; number of 3-team cycles A beats B beats C beats A | Combinatorics | Counting | 22 |
| 2016 AMC 10B | In regular hexagon ABCDEF, points W,X,Y,Z on sides; lines AB,ZW,YX,ED parallel and equally spaced; find area ratio of WCXYFZ to entire hexagon | Geometry | Triangles & Polygons | 23 |
| 2016 AMC 10B | Four-digit integers abcd where two-digit numbers ab, bc, cd form increasing arithmetic sequence; count how many | Algebra | Sequences & Series | 24 |
| 2016 AMC 10B | f(x)=sum_{k=2}^{10} (floor(kx)-k floor(x)), count distinct values for x≥0 | Algebra | Functions | 25 |
| 2015 AMC 10A | Simplify (2^0-1+5^2-0)^{-1} times 5 | Algebra | Logarithms & Exponents | 1 |
| 2015 AMC 10A | Box of 25 triangular and square tiles with 84 edges, find number of square tiles | Algebra | Rates, Mixtures, & Averages | 2 |
| 2015 AMC 10A | Add toothpicks to extend a 3-step staircase to a 5-step staircase | Algebra | Sequences & Series | 3 |
| 2015 AMC 10A | Equalize eggs; Pablo 3x Sofia, Sofia 2x Mia; fraction of Pablo's eggs to Sofia | Algebra | Rates, Mixtures, & Averages | 4 |
| 2015 AMC 10A | Class average 80 without Payton, 81 with; find Payton's score | Algebra | Rates, Mixtures, & Averages | 5 |
| 2015 AMC 10A | Sum of two numbers is 5 times their difference; find ratio of larger to smaller | Algebra | Systems & Algebraic Manipulation | 6 |
| 2015 AMC 10A | Number of terms in arithmetic sequence 13, 16, 19, ..., 73 | Algebra | Sequences & Series | 7 |
| 2015 AMC 10A | Find years until Pete is twice Claire, given ages two and four years ago | Algebra | Rates, Mixtures, & Averages | 8 |
| 2015 AMC 10A | Two cylinders have equal volume, second radius 10% larger; compare heights | Algebra | Systems & Algebraic Manipulation | 9 |
| 2015 AMC 10A | Rearrangements of abcd with no two adjacent letters also adjacent in alphabet | Combinatorics | Counting | 10 |
| 2015 AMC 10A | Rectangle length:width=4:3, diagonal d, area kd^2; find k | Geometry | Triangles & Polygons | 11 |
| 2015 AMC 10A | Given points (√π, a) and (√π, b) on x^4+y^2=2x^2y+1, find |a-b| | Algebra | Systems & Algebraic Manipulation | 12 |
| 2015 AMC 10A | 12 coins each 5¢ or 10¢, exactly 17 different totals; number of 10¢ coins | Combinatorics | Counting | 13 |
| 2015 AMC 10A | Disk rolls clockwise around clock; when does arrow next point upward? | Geometry | Circular Geometry | 14 |
| 2015 AMC 10A | Relatively prime fraction increases by 10% when both terms increased by 1 | Algebra | Systems & Algebraic Manipulation | 15 |
| 2015 AMC 10A | Solve symmetric system y+4=(x-2)^2, x+4=(y-2)^2 with x≠y; find x^2+y^2 | Algebra | Systems & Algebraic Manipulation | 16 |
| 2015 AMC 10A | Equilateral triangle from x=1, y=1+√3/3 x, and a line through origin; perimeter | Geometry | Coordinate Geometry | 17 |
| 2015 AMC 10A | Among first 1000 integers, count those with only numeric digits in hexadecimal | Combinatorics | Counting | 18 |
| 2015 AMC 10A | Isosceles right triangle area 12.5, trisect right angle; area of middle triangle | Geometry | Triangles & Polygons | 19 |
| 2015 AMC 10A | Rectangle integer sides; which of 100,102,104,106,108 cannot equal area+perimeter? | Number Theory | Factors, Multiples & Divisibility | 20 |
| 2015 AMC 10A | Tetrahedron with AB=5, AC=3, BC=4, BD=4, AD=3, CD=12√2/5; find volume | Geometry | 3D Geometry & Solids | 21 |
| 2015 AMC 10A | 8 people flip fair coins; probability no two adjacent people stand | Combinatorics | Probability | 22 |
| 2015 AMC 10A | Zeros of x^2-ax+2a are integers; sum of possible values of a | Algebra | Polynomials | 23 |
| 2015 AMC 10A | Quadrilateral with AB=2, right angles B,C, CD=AD, integer sides, count p<2015 | Geometry | Triangles & Polygons | 24 |
| 2015 AMC 10A | Two random points on sides of unit square; probability distance is at least 1/2 | Combinatorics | Probability | 25 |
| 2015 AMC 10B | Evaluate 2 minus (-2) to the -2 power | Algebra | Logarithms & Exponents | 1 |
| 2015 AMC 10B | Marie finishes three equal tasks, find finish time of third task | Algebra | Rates, Mixtures, & Averages | 2 |
| 2015 AMC 10B | Sum of two numbers twice and three times is 100, one is 28, find the other | Algebra | Systems & Algebraic Manipulation | 3 |
| 2015 AMC 10B | Four siblings eat fractions of a pizza, order by amount consumed | Algebra | Rates, Mixtures, & Averages | 4 |
| 2015 AMC 10B | Six-person race with position constraints, find who finished 8th | Algebra | Logic & Truth Values | 5 |
| 2015 AMC 10B | Mahdi's sport schedule with constraints, find the day he swims | Algebra | Logic & Truth Values | 6 |
| 2015 AMC 10B | Evaluate nested custom operation a diamond b = a - 1/b | Algebra | Functions | 7 |
| 2015 AMC 10B | Letter F rotated 90° clockwise, reflected in y-axis, half turn; find final image | Geometry | Coordinate Geometry | 8 |
| 2015 AMC 10B | Area of shark's fin shape bounded by quarter-circles and line segment | Geometry | Circular Geometry | 9 |
| 2015 AMC 10B | Product of odd negative integers > -2015, find sign and units digit | Number Theory | Modular Arithmetic | 10 |
| 2015 AMC 10B | Probability a number with prime digits is prime, selected from numbers less than 100 | Combinatorics | Probability | 11 |
| 2015 AMC 10B | Integer points (x,-x) inside or on circle radius 10 centered at (5,5) | Geometry | Coordinate Geometry | 12 |
| 2015 AMC 10B | Sum of altitudes of triangle formed by line 12x+5y=60 and axes | Geometry | Coordinate Geometry | 13 |
| 2015 AMC 10B | Maximize sum of roots of (x-a)(x-b)+(x-b)(x-c)=0 for distinct one-digit a,b,c | Algebra | Polynomials | 14 |
| 2015 AMC 10B | Given ratios of people, horses, sheep, cows, ducks, find impossible total count | Number Theory | Factors, Multiples & Divisibility | 15 |
| 2015 AMC 10B | Probability Al, Bill, Cal numbers are multiples with distinct assignments 1-10 | Combinatorics | Probability | 16 |
| 2015 AMC 10B | Volume of octahedron formed by joining face centers of 3x4x5 prism | Geometry | 3D Geometry & Solids | 17 |
| 2015 AMC 10B | Expected number of heads after flipping coins and retossing tails up to three times | Combinatorics | Probability | 18 |
| 2015 AMC 10B | Right triangle ABC with squares, X,Y,Z,W concyclic, find perimeter | Geometry | Circular Geometry | 19 |
| 2015 AMC 10B | Ant on cube visits all corners exactly once in 7 moves, can't return; count paths | Combinatorics | Counting | 20 |
| 2015 AMC 10B | Staircase with jumps of 2 and 5 steps, Dash takes 19 fewer jumps than Cozy | Number Theory | Modular Arithmetic | 21 |
| 2015 AMC 10B | Regular pentagon with diagonals, AG=1, find FG+JH+CD | Geometry | Triangles & Polygons | 22 |
| 2015 AMC 10B | n! ends in k zeros, (2n)! ends in 3k zeros, find sum of four smallest n | Number Theory | Factors, Multiples & Divisibility | 23 |
| 2015 AMC 10B | Ant walks spiral path on coordinate plane, find p_2015 | Algebra | Sequences & Series | 24 |
| 2015 AMC 10B | Rectangular box with integer dimensions, volume equals surface area, count ordered triples | Algebra | Systems & Algebraic Manipulation | 25 |
| 2014 AMC 10A | Evaluate 10*(1/2+1/5+1/10)^{-1} and simplify. | Algebra | Systems & Algebraic Manipulation | 1 |
| 2014 AMC 10A | Cat eats 1/3 can in morning and 1/4 in evening; after how many days is 6-can box finished starting Monday morning? | Algebra | Rates, Mixtures, & Averages | 2 |
| 2014 AMC 10A | Bridget sells 48 loaves at different prices during the day; find total profit given cost per loaf. | Algebra | Rates, Mixtures, & Averages | 3 |
| 2014 AMC 10A | Four painted houses in a row with color order constraints; count possible arrangements. | Combinatorics | Counting | 4 |
| 2014 AMC 10A | Percentages of quiz scores given; find difference between mean and median. | Algebra | Rates, Mixtures, & Averages | 5 |
| 2014 AMC 10A | a cows give b gallons in c days; find gallons from d cows in e days. | Algebra | Rates, Mixtures, & Averages | 6 |
| 2014 AMC 10A | Given x < a and y < b, determine how many inequalities must be true. | Algebra | Inequalities | 7 |
| 2014 AMC 10A | Which of five factorial expressions is a perfect square? | Number Theory | Factors, Multiples & Divisibility | 8 |
| 2014 AMC 10A | Right triangle legs 2√3 and 6; find length of third altitude. | Geometry | Triangles & Polygons | 9 |
| 2014 AMC 10A | Five positive consecutive integers starting with a have average b; find average of five integers starting with b. | Algebra | Sequences & Series | 10 |
| 2014 AMC 10A | Customer has three coupons; for which listed price does coupon 1 give the greatest discount? | Algebra | Inequalities | 11 |
| 2014 AMC 10A | Regular hexagon side 6 with six 120° sectors of radius 3; find shaded area inside hexagon. | Geometry | Circular Geometry | 12 |
| 2014 AMC 10A | Equilateral triangle side 1 with outward squares on each side; find area of hexagon formed. | Geometry | Triangles & Polygons | 13 |
| 2014 AMC 10A | Two perpendicular lines intersect at (6,8) and have y-intercepts summing to zero; find area of triangle formed. | Geometry | Coordinate Geometry | 14 |
| 2014 AMC 10A | David drives 35 miles first hour, then speeds up; arrives 30 min early instead of 1 h late; find total distance. | Algebra | Rates, Mixtures, & Averages | 15 |
| 2014 AMC 10A | Rectangle ABCD with midpoints; shade region and find its area. | Geometry | Triangles & Polygons | 16 |
| 2014 AMC 10A | Three fair dice rolled; probability that two numbers sum to the third number. | Combinatorics | Probability | 17 |
| 2014 AMC 10A | Square has y-coordinates 0,1,4,5; find its area. | Geometry | Coordinate Geometry | 18 |
| 2014 AMC 10A | Stacked cubes of edge 1,2,3,4; find length of segment XY inside cube 3. | Geometry | 3D Geometry & Solids | 19 |
| 2014 AMC 10A | Product 8 * (888...8) with k digits has digit sum 1000; find k. | Number Theory | Modular Arithmetic | 20 |
| 2014 AMC 10A | Lines y=ax+5 and y=3x+b intersect x-axis at same point; a,b positive integers; find sum of all possible x-coordinates. | Algebra | Systems & Algebraic Manipulation | 21 |
| 2014 AMC 10A | Rectangle AB=20, BC=10, E on CD with angle CBE=15°; find AE. | Geometry | Triangles & Polygons | 22 |
| 2014 AMC 10A | Rectangular paper length=√3 times width, folded along dotted line; find ratio B/A of folded area to original. | Geometry | Triangles & Polygons | 23 |
| 2014 AMC 10A | Sequence built by listing n+3 numbers then skipping n; find 500,000th term. | Algebra | Sequences & Series | 24 |
| 2014 AMC 10A | Given 5^867 between 2^2013 and 2^2014; count pairs (m,n) with 1≤m≤2012 satisfying 5^n<2^m<2^{m+2}<5^{n+1}. | Algebra | Logarithms & Exponents | 25 |
| 2014 AMC 10B | Leah has 13 coins; with one more nickel, pennies equal nickels. Find total value. | Algebra | Rates, Mixtures, & Averages | 1 |
| 2014 AMC 10B | Simplify (2^3+2^3)/(2^{-3}+2^{-3}) to a number. | Algebra | Logarithms & Exponents | 2 |
| 2014 AMC 10B | Randy drives first third on gravel, 20 miles on pavement, one-fifth on dirt. Find total trip length. | Algebra | Rates, Mixtures, & Averages | 3 |
| 2014 AMC 10B | Susie buys 4 muffins and 3 bananas; Calvin buys 2 muffins and 16 bananas for twice the cost. Find muffin-to-banana cost ratio. | Algebra | Systems & Algebraic Manipulation | 4 |
| 2014 AMC 10B | A square window made of 8 equal panes (5:2 ratio) and 2-inch borders. Find the side length of the window. | Algebra | Systems & Algebraic Manipulation | 5 |
| 2014 AMC 10B | Orvin can buy 30 balloons normally; with 'buy one, second 1/3 off', how many can he buy? | Algebra | Rates, Mixtures, & Averages | 6 |
| 2014 AMC 10B | If A > B > 0 and A is x% greater than B, express x in terms of A and B. | Algebra | Systems & Algebraic Manipulation | 7 |
| 2014 AMC 10B | A truck travels b/6 feet every t seconds. How many yards in 3 minutes? | Algebra | Rates, Mixtures, & Averages | 8 |
| 2014 AMC 10B | Given a complex fraction equals 2014, find (w+z)/(w-z). | Algebra | Systems & Algebraic Manipulation | 9 |
| 2014 AMC 10B | Cryptarithm: ABBCB+BCADA=DBDDD with distinct digits; how many possible D values? | Number Theory | Modular Arithmetic | 10 |
| 2014 AMC 10B | Find smallest integer n% single discount better than two 15% off, three 10% off, or 25% then 5% off. | Algebra | Rates, Mixtures, & Averages | 11 |
| 2014 AMC 10B | Find the fifth-largest divisor of 2,014,000,000. | Number Theory | Factors, Multiples & Divisibility | 12 |
| 2014 AMC 10B | Six regular hexagons surround a hexagon of side 1; find area of triangle ABC formed. | Geometry | Triangles & Polygons | 13 |
| 2014 AMC 10B | Danica drives whole hours at 55 mph; odometer start abc end cba with digit sum ≤7. Find a^2+b^2+c^2. | Number Theory | Modular Arithmetic | 14 |
| 2014 AMC 10B | Rectangle with DC=2BC, E,F trisect angle ADC; find ratio of area triangle DEF to rectangle ABCD. | Geometry | Triangles & Polygons | 15 |
| 2014 AMC 10B | Four fair dice rolled; probability at least three show same value. | Combinatorics | Probability | 16 |
| 2014 AMC 10B | Find the greatest power of 2 dividing 10^{1002} - 4^{501}. | Number Theory | Factors, Multiples & Divisibility | 17 |
| 2014 AMC 10B | 11 positive integers, mean 10, median 9, unique mode 8; maximize the largest integer. | Algebra | Rates, Mixtures, & Averages | 18 |
| 2014 AMC 10B | Two concentric circles radii 1 and 2; two random points on outer circle; probability chord intersects inner circle. | Geometry | Circular Geometry | 19 |
| 2014 AMC 10B | How many integers x make x^4-51x^2+50 negative? | Algebra | Inequalities | 20 |
| 2014 AMC 10B | Trapezoid ABCD, AB||CD, AB=33, CD=21, legs 10 and 14; acute A and B; find shorter diagonal. | Geometry | Triangles & Polygons | 21 |
| 2014 AMC 10B | Eight semicircles inside a square of side 2; find radius of circle tangent to all. | Geometry | Circular Geometry | 22 |
| 2014 AMC 10B | Sphere inscribed in truncated cone; volume of frustum twice sphere; find ratio of bottom to top radius. | Geometry | 3D Geometry & Solids | 23 |
| 2014 AMC 10B | Numbers 1-5 in a circle; arrangement is bad if not all sums 1-15 achievable by consecutive numbers. Count bad arrangements (mod rotation/reflection). | Combinatorics | Counting | 24 |
| 2014 AMC 10B | Frog on lily pads 1, escapes to 10, eaten at 0; transition probabilities based on pad. Start pad 1. Find escape probability. | Combinatorics | Probability | 25 |
| 2013 AMC 10A | Find cost of a 5-mile taxi ride given base fare and per-mile rate. | Algebra | Rates, Mixtures, & Averages | 1 |
| 2013 AMC 10A | How many 1/4-cup measures to make 2 1/2 cups of sugar? | Algebra | Rates, Mixtures, & Averages | 2 |
| 2013 AMC 10A | Square of side 10, point E on BC, area of triangle ABE = 40; find BE. | Geometry | Triangles & Polygons | 3 |
| 2013 AMC 10A | Softball team scores 1..10 runs; deduce opponent runs and find total. | Algebra | Systems & Algebraic Manipulation | 4 |
| 2013 AMC 10A | Split vacation costs evenly; find difference t-d after reimbursements. | Algebra | Rates, Mixtures, & Averages | 5 |
| 2013 AMC 10A | Deduce Kris's age from sibling activities and age constraints. | Algebra | Logic & Truth Values | 6 |
| 2013 AMC 10A | Choose 4 courses with English and at least one math; count ways. | Combinatorics | Counting | 7 |
| 2013 AMC 10A | Simplify fraction with large powers of 2. | Algebra | Logarithms & Exponents | 8 |
| 2013 AMC 10A | Basketball: 20% of 3-pt, 30% of 2-pt successful on 30 attempts; points? | Algebra | Rates, Mixtures, & Averages | 9 |
| 2013 AMC 10A | Bouquet of roses and carnations; given fractions, find percent carnations. | Algebra | Rates, Mixtures, & Averages | 10 |
| 2013 AMC 10A | Council: 10 ways to choose 2-person committee; how many for 3-person? | Combinatorics | Counting | 11 |
| 2013 AMC 10A | Isosceles triangle with parallel lines; find perimeter of parallelogram. | Geometry | Triangles & Polygons | 12 |
| 2013 AMC 10A | Count three-digit numbers not divisible by 5, digit sum <20, first=last digit. | Combinatorics | Counting | 13 |
| 2013 AMC 10A | Solid cube 3x3x3 with unit cubes removed from each corner; count edges. | Geometry | 3D Geometry & Solids | 14 |
| 2013 AMC 10A | Triangle sides 10, 15; altitude to third side is average of other altitudes. | Geometry | Triangles & Polygons | 15 |
| 2013 AMC 10A | Triangle reflected across x=8; find area of union of original and reflected. | Geometry | Coordinate Geometry | 16 |
| 2013 AMC 10A | Alice, Beatrix, Claire visit every 3,4,5 days; count days exactly two visit. | Number Theory | Factors, Multiples & Divisibility | 17 |
| 2013 AMC 10A | Quadrilateral cut into equal-area halves by line through A; find intersection. | Geometry | Coordinate Geometry | 18 |
| 2013 AMC 10A | Count bases b>3 such that 2013 ends in digit 3 in base b. | Number Theory | Modular Arithmetic | 19 |
| 2013 AMC 10A | Area swept out by rotating a unit square 45° about its center. | Geometry | Circular Geometry | 20 |
| 2013 AMC 10A | Pirates take fraction of remaining gold; smallest initial to give integer shares. | Algebra | Sequences & Series | 21 |
| 2013 AMC 10A | Six unit spheres at hexagon vertices inside larger sphere; find 8th sphere radius. | Geometry | 3D Geometry & Solids | 22 |
| 2013 AMC 10A | Triangle ABC, circle center A radius AB meets BC at X; integer BX, CX; find BC. | Geometry | Circular Geometry | 23 |
| 2013 AMC 10A | Schedule 6 rounds of 3-on-3 backgammon with each pair playing twice; count ways. | Combinatorics | Counting | 24 |
| 2013 AMC 10A | All diagonals drawn in a regular octagon; count interior intersection points. | Combinatorics | Counting | 25 |
| 2013 AMC 10B | Evaluate the expression (2+4+6)/(1+3+5) - (1+3+5)/(2+4+6) | Algebra | Systems & Algebraic Manipulation | 1 |
| 2013 AMC 10B | Find pounds of potatoes from garden based on step measurements | Algebra | Rates, Mixtures, & Averages | 2 |
| 2013 AMC 10B | Find low temperature given high is 16 more and average is 3 | Algebra | Rates, Mixtures, & Averages | 3 |
| 2013 AMC 10B | Determine the nth number when counting backwards from 201 to 3 | Algebra | Sequences & Series | 4 |
| 2013 AMC 10B | Minimize 2a - ab given a,b positive integers less than 6 | Algebra | Systems & Algebraic Manipulation | 5 |
| 2013 AMC 10B | Find overall average age of fifth-graders and parents | Algebra | Rates, Mixtures, & Averages | 6 |
| 2013 AMC 10B | Find area of triangle formed by three equally spaced points on a circle | Geometry | Triangles & Polygons | 7 |
| 2013 AMC 10B | Find combined miles per gallon for two cars driving same distance | Algebra | Rates, Mixtures, & Averages | 8 |
| 2013 AMC 10B | Sum of three pairwise relatively prime integers with product 27000 | Number Theory | Factors, Multiples & Divisibility | 9 |
| 2013 AMC 10B | Find number of three-point shots attempted given percentages and totals | Algebra | Rates, Mixtures, & Averages | 10 |
| 2013 AMC 10B | Find x+y satisfying x^2+y^2=10x-6y-34 | Algebra | Systems & Algebraic Manipulation | 11 |
| 2013 AMC 10B | Probability two chosen segments from a regular pentagon have same length | Combinatorics | Probability | 12 |
| 2013 AMC 10B | Find the 53rd number spoken in a turn-taking counting game | Algebra | Sequences & Series | 13 |
| 2013 AMC 10B | Describe set of points where a custom operation is commutative | Algebra | Functions | 14 |
| 2013 AMC 10B | Find ratio a/b when equilateral triangle and regular hexagon have equal area | Geometry | Triangles & Polygons | 15 |
| 2013 AMC 10B | Find area of quadrilateral AECD given medians and segment lengths in triangle ABC | Geometry | Triangles & Polygons | 16 |
| 2013 AMC 10B | Find maximum silver tokens from exchange booths with red and blue tokens | Algebra | Systems & Algebraic Manipulation | 17 |
| 2013 AMC 10B | Count integers 1000-2013 where units digit equals sum of other digits | Combinatorics | Counting | 18 |
| 2013 AMC 10B | Find the double root of quadratic with coefficients in arithmetic progression | Algebra | Polynomials | 19 |
| 2013 AMC 10B | Find |a1-b1| when 2013 expressed as ratio of factorials with minimal a1+b1 | Number Theory | Factors, Multiples & Divisibility | 20 |
| 2013 AMC 10B | Smallest N from two sequences with same seventh term | Algebra | Sequences & Series | 21 |
| 2013 AMC 10B | Number of ways to assign digits to octagon vertices and center with equal line sums | Combinatorics | Counting | 22 |
| 2013 AMC 10B | Find length DF in triangle with perpendiculars and given side lengths | Geometry | Triangles & Polygons | 23 |
| 2013 AMC 10B | How many numbers 2010-2019 are 'nice' (sum of four divisors of m) | Number Theory | Factors, Multiples & Divisibility | 24 |
| 2013 AMC 10B | Count N with last two digits of base-5 and base-6 sum matching 2N | Number Theory | Modular Arithmetic | 25 |
| 2012 AMC 10A | Cagney frosts a cupcake every 20 s, Lacey every 30 s; together how many in 5 min? | Algebra | Rates, Mixtures, & Averages | 1 |
| 2012 AMC 10A | A square of side 8 is cut in half; find dimensions of a rectangle. | Geometry | Triangles & Polygons | 2 |
| 2012 AMC 10A | Bug crawls from -2 to -6 to 5 on number line; total units crawled. | Algebra | Rates, Mixtures, & Averages | 3 |
| 2012 AMC 10A | Angles 24° and 20° sharing ray; smallest possible third angle. | Geometry | Triangles & Polygons | 4 |
| 2012 AMC 10A | 100 adult cats, half female, half have litters of 4 kittens; total cats and kittens. | Algebra | Rates, Mixtures, & Averages | 5 |
| 2012 AMC 10A | Two positive numbers product 9, one reciprocal is 4 times other; find sum. | Algebra | Systems & Algebraic Manipulation | 6 |
| 2012 AMC 10A | Bag 3/5 blue, rest red, double red; fraction red after doubling. | Algebra | Rates, Mixtures, & Averages | 7 |
| 2012 AMC 10A | Three numbers pairwise sums 12,17,19; find the middle number. | Algebra | Systems & Algebraic Manipulation | 8 |
| 2012 AMC 10A | Dice: one even faces only, one odd faces only; probability sum is 7. | Combinatorics | Probability | 9 |
| 2012 AMC 10A | Circle split into 12 sectors with integer angles in arithmetic sequence; smallest angle. | Algebra | Sequences & Series | 10 |
| 2012 AMC 10A | Two externally tangent circles radii 5 and 3, common external tangent; find BC. | Geometry | Circular Geometry | 11 |
| 2012 AMC 10A | Leap year rules; 200th birthday Tuesday, find day of week born. | Number Theory | Modular Arithmetic | 12 |
| 2012 AMC 10A | Iterative average of 1,2,3,4,5; difference between max and min possible values. | Algebra | Rates, Mixtures, & Averages | 13 |
| 2012 AMC 10A | 31x31 checkerboard black corners; how many black squares? | Combinatorics | Counting | 14 |
| 2012 AMC 10A | Three unit squares and two segments connecting vertices; area of triangle ABC. | Geometry | Triangles & Polygons | 15 |
| 2012 AMC 10A | Three runners on 500 m track speeds 4.4,4.8,5.0; time until all together. | Algebra | Rates, Mixtures, & Averages | 16 |
| 2012 AMC 10A | Relatively prime a,b with fraction condition; find a-b. | Algebra | Polynomials | 17 |
| 2012 AMC 10A | Closed curve of 9 arcs from regular hexagon centers; enclosed area. | Geometry | Circular Geometry | 18 |
| 2012 AMC 10A | Paula and two helpers paint house with lunch break; break length in minutes. | Algebra | Rates, Mixtures, & Averages | 19 |
| 2012 AMC 10A | 3x3 grid random colors, rotate 90° and recolor; probability all black. | Combinatorics | Probability | 20 |
| 2012 AMC 10A | Points (0,0,0),(1,0,0),(0,2,0),(0,0,3); midpoints form quadrilateral area. | Geometry | 3D Geometry & Solids | 21 |
| 2012 AMC 10A | Sum of first m odd integers is 212 more than sum of first n even; sum of all possible n. | Algebra | Sequences & Series | 22 |
| 2012 AMC 10A | Six students, each same number of friends; total number of friendship arrangements. | Combinatorics | Counting | 23 |
| 2012 AMC 10A | Integers a≥b≥c>0 satisfying two given equations; find a. | Algebra | Systems & Algebraic Manipulation | 24 |
| 2012 AMC 10A | Real numbers x,y,z in [0,n]; probability none within 1 > 1/2, smallest integer n. | Combinatorics | Probability | 25 |
| 2012 AMC 10B | Students and rabbits per classroom, total difference across four rooms | Algebra | Rates, Mixtures, & Averages | 1 |
| 2012 AMC 10B | Circle inscribed in a 2:1 rectangle, find area of rectangle | Geometry | Circular Geometry | 2 |
| 2012 AMC 10B | Reflect point (1000,2012) across y=2000, find new coordinates | Geometry | Coordinate Geometry | 3 |
| 2012 AMC 10B | Leftover marbles after packing 6 per bag, find combined leftovers | Number Theory | Modular Arithmetic | 4 |
| 2012 AMC 10B | Dinner cost with 10% tax and 15% tip on pre-tax, find original price | Algebra | Rates, Mixtures, & Averages | 5 |
| 2012 AMC 10B | Effect of rounding x up and y down on estimate of x-y | Algebra | Systems & Algebraic Manipulation | 6 |
| 2012 AMC 10B | Animals hiding acorns, different per hole, same total, find number | Algebra | Rates, Mixtures, & Averages | 7 |
| 2012 AMC 10B | Sum of integer solutions to compound quadratic inequality | Algebra | Inequalities | 8 |
| 2012 AMC 10B | Minimum odd integers among six with given consecutive sums | Number Theory | Modular Arithmetic | 9 |
| 2012 AMC 10B | Count ordered pairs of positive integers solving M/6 = 6/N | Number Theory | Factors, Multiples & Divisibility | 10 |
| 2012 AMC 10B | Dessert menus for a week with no repeat adjacents and cake on Friday | Combinatorics | Counting | 11 |
| 2012 AMC 10B | Points A, B, C, D with directions and distances, estimate AD | Geometry | Triangles & Polygons | 12 |
| 2012 AMC 10B | Escalator: walking vs standing times, find ride-down time | Algebra | Rates, Mixtures, & Averages | 13 |
| 2012 AMC 10B | Two equilateral triangles in a square overlap as a rhombus, find area | Geometry | Triangles & Polygons | 14 |
| 2012 AMC 10B | Round-robin 6 teams, maximum teams tied for most wins | Combinatorics | Counting | 15 |
| 2012 AMC 10B | Three mutually tangent circles radius 2, total area of circles and inner region | Geometry | Circular Geometry | 16 |
| 2012 AMC 10B | Paper disk cut into two sectors forming cones, volume ratio smaller to larger | Geometry | 3D Geometry & Solids | 17 |
| 2012 AMC 10B | False positive test probability, find P(disease | positive test) | Combinatorics | Probability | 18 |
| 2012 AMC 10B | Rectangle with extended side, intersection and quadrilateral area | Geometry | Triangles & Polygons | 19 |
| 2012 AMC 10B | Game of doubling and adding 50, smallest starting number for Bernardo to win | Algebra | Sequences & Series | 20 |
| 2012 AMC 10B | Four points, segment lengths a,a,a,a,2a,b, find b:a ratio | Geometry | Triangles & Polygons | 21 |
| 2012 AMC 10B | List of first 10 integers with adjacent-preview condition, count lists | Combinatorics | Counting | 22 |
| 2012 AMC 10B | Tetrahedron cut from unit cube, remaining shape height on face | Geometry | 3D Geometry & Solids | 23 |
| 2012 AMC 10B | Three girls like four songs with pairwise-disjoint-like constraints, count ways | Combinatorics | Counting | 24 |
| 2012 AMC 10B | Bug traverses hexagonal lattice with direction arrows, count paths | Combinatorics | Counting | 25 |
| 2011 AMC 10A | Cell phone plan cost with texts and overage minutes | Algebra | Rates, Mixtures, & Averages | 1 |
| 2011 AMC 10A | Minimum small bottles to fill a large bottle | Algebra | Rates, Mixtures, & Averages | 2 |
| 2011 AMC 10A | Evaluate nested custom average operations | Algebra | Functions | 3 |
| 2011 AMC 10A | Difference of two arithmetic sequence sums X and Y | Algebra | Sequences & Series | 4 |
| 2011 AMC 10A | Weighted average of running times with grade ratios | Algebra | Rates, Mixtures, & Averages | 5 |
| 2011 AMC 10A | Smallest possible size of union of two sets | Combinatorics | Counting | 6 |
| 2011 AMC 10A | Identify the equation with no solution | Algebra | Functions | 7 |
| 2011 AMC 10A | Percentage of geese among non-swans birds | Algebra | Rates, Mixtures, & Averages | 8 |
| 2011 AMC 10A | Area of rectangle bounded by given line equations | Geometry | Coordinate Geometry | 9 |
| 2011 AMC 10A | Cost of pencil in cents given total cost and constraints | Number Theory | Factors, Multiples & Divisibility | 10 |
| 2011 AMC 10A | Ratio of areas of inscribed square to outer square | Geometry | Triangles & Polygons | 11 |
| 2011 AMC 10A | Number of free throws given basketball scoring conditions | Algebra | Rates, Mixtures, & Averages | 12 |
| 2011 AMC 10A | Count even three-digit numbers with distinct given digits | Combinatorics | Counting | 13 |
| 2011 AMC 10A | Probability circle area < circumference from dice sum | Combinatorics | Probability | 14 |
| 2011 AMC 10A | Total trip miles given battery and gasoline usage rates | Algebra | Rates, Mixtures, & Averages | 15 |
| 2011 AMC 10A | Simplify sum of nested square roots with radicals | Algebra | Systems & Algebraic Manipulation | 16 |
| 2011 AMC 10A | Find A+H in sequence with constant sum of consecutive terms | Algebra | Sequences & Series | 17 |
| 2011 AMC 10A | Area inside one circle but outside two others in a configuration | Geometry | Circular Geometry | 18 |
| 2011 AMC 10A | Percent population growth over 20 years given perfect square conditions | Number Theory | Factors, Multiples & Divisibility | 19 |
| 2011 AMC 10A | Probability that two clockwise chords of length r on a circle intersect | Combinatorics | Probability | 20 |
| 2011 AMC 10A | Probability all four selected coins are genuine given equal weight pairs | Combinatorics | Probability | 21 |
| 2011 AMC 10A | Number of colorings of pentagon vertices with diagonal color restrictions | Combinatorics | Counting | 22 |
| 2011 AMC 10A | Find the last number said after sequential skipping patterns | Number Theory | Modular Arithmetic | 23 |
| 2011 AMC 10A | Volume of intersection of two regular tetrahedra inside a unit cube | Geometry | 3D Geometry & Solids | 24 |
| 2011 AMC 10A | Count points dividing unit square into 100 equal-area triangles but not 60 | Combinatorics | Counting | 25 |
| 2011 AMC 10B | Simplify the difference of two fractions: (2+4+6)/(1+3+5) minus its reciprocal. | Algebra | Systems & Algebraic Manipulation | 1 |
| 2011 AMC 10B | Find minimum next test score to raise average by 3 points from given scores. | Algebra | Rates, Mixtures, & Averages | 2 |
| 2011 AMC 10B | Rectangular tile dimensions reported to nearest inch, find minimum possible area. | Algebra | Systems & Algebraic Manipulation | 3 |
| 2011 AMC 10B | Two people share trip costs; how much must LeRoy give Bernardo to equalize? | Algebra | Rates, Mixtures, & Averages | 4 |
| 2011 AMC 10B | Reversed digits of two-digit a gave product 161; find correct product of a and b. | Number Theory | Factors, Multiples & Divisibility | 5 |
| 2011 AMC 10B | Casper eats fractions of candies and gives some away over three days; find starting amount. | Algebra | Systems & Algebraic Manipulation | 6 |
| 2011 AMC 10B | Triangle's two angles sum to 108° and differ by 30°; find the largest angle. | Geometry | Triangles & Polygons | 7 |
| 2011 AMC 10B | Beach crowded if at least 80°F and sunny; it's not crowded; what can be concluded? | Algebra | Logic & Truth Values | 8 |
| 2011 AMC 10B | Right triangle ABC, area of triangle EBD is one third of ABC; find BD. | Geometry | Triangles & Polygons | 9 |
| 2011 AMC 10B | Set of powers of 10 up to 10^10; ratio of largest to sum of others closest to which integer? | Algebra | Sequences & Series | 10 |
| 2011 AMC 10B | 52 people, 12 months; largest n such that at least n have same birth month always true. | Combinatorics | Counting | 11 |
| 2011 AMC 10B | Track with straight sides and semicircular ends, width 6 m, 36 s longer outside; find speed. | Algebra | Rates, Mixtures, & Averages | 12 |
| 2011 AMC 10B | Two real numbers chosen from [-20,10]; probability their product is positive. | Combinatorics | Probability | 13 |
| 2011 AMC 10B | Rectangle with diagonal 25 and area 168; find its perimeter. | Algebra | Systems & Algebraic Manipulation | 14 |
| 2011 AMC 10B | Custom operation a@b = (a+b)/2; which distributive laws hold for all numbers? | Algebra | Functions | 15 |
| 2011 AMC 10B | Dart board regular octagon; probability dart lands in center square. | Geometry | Triangles & Polygons | 16 |
| 2011 AMC 10B | Circle with diameter EB parallel to DC; given angle ratio, find angle BCD. | Geometry | Circular Geometry | 17 |
| 2011 AMC 10B | Rectangle ABCD, AB=6, BC=3, M on AB with ∠AMD = ∠CMD; find ∠AMD. | Geometry | Triangles & Polygons | 18 |
| 2011 AMC 10B | Solve √(5|x|+8) = √(x^2-16) and find product of all real roots. | Algebra | Systems & Algebraic Manipulation | 19 |
| 2011 AMC 10B | Rhombus side 2, ∠B=120°; region R closer to B than other vertices; find area. | Geometry | Triangles & Polygons | 20 |
| 2011 AMC 10B | Four integers w>x>y>z sum to 44, six pairwise positive differences given; sum of possible w. | Algebra | Systems & Algebraic Manipulation | 21 |
| 2011 AMC 10B | Pyramid square base side 1, lateral faces equilateral; inscribed cube; find its volume. | Geometry | 3D Geometry & Solids | 22 |
| 2011 AMC 10B | Find the hundreds digit of 2011^2011. | Number Theory | Modular Arithmetic | 23 |
| 2011 AMC 10B | Line y=mx+2 passes through no lattice point for 0<x≤100; max a where 1/2<m<a. | Number Theory | Factors, Multiples & Divisibility | 24 |
| 2011 AMC 10B | Sequence of triangles from incircle tangency points; perimeter of last valid triangle. | Geometry | Triangles & Polygons | 25 |
| 2010 AMC 10A | Find average width of five books | Algebra | Rates, Mixtures, & Averages | 1 |
| 2010 AMC 10A | Four squares and a rectangle form a large square; find length-to-width ratio of rectangle | Geometry | Triangles & Polygons | 2 |
| 2010 AMC 10A | Tyrone gives marbles to Eric so he has twice as many; find marbles given | Algebra | Rates, Mixtures, & Averages | 3 |
| 2010 AMC 10A | Book recorded onto equal-length discs; find minutes per disc using minimum discs | Algebra | Rates, Mixtures, & Averages | 4 |
| 2010 AMC 10A | Circle circumference 24π; find area coefficient k | Geometry | Circular Geometry | 5 |
| 2010 AMC 10A | Evaluate custom operation ♠(2, ♠(2,2)) defined as x - 1/y | Algebra | Functions | 6 |
| 2010 AMC 10A | Crystal runs north, northeast, southeast, then straight back; find return distance | Geometry | Coordinate Geometry | 7 |
| 2010 AMC 10A | Tony’s pay based on age; worked 50 days earned $630; find his age at end | Algebra | Rates, Mixtures, & Averages | 8 |
| 2010 AMC 10A | Three-digit palindrome x where x+32 is a four-digit palindrome; sum of digits | Algebra | Systems & Algebraic Manipulation | 9 |
| 2010 AMC 10A | Marvin's birthday Tuesday May 27, 2008; next year it falls on Saturday | Number Theory | Modular Arithmetic | 10 |
| 2010 AMC 10A | Inequality a ≤ 2x+3 ≤ b has solution interval length 10; find b-a | Algebra | Inequalities | 11 |
| 2010 AMC 10A | Scale model water tower; find model height given volume ratio | Geometry | 3D Geometry & Solids | 12 |
| 2010 AMC 10A | Angelina drove at two speeds with a gas stop; identify correct distance equation | Algebra | Rates, Mixtures, & Averages | 13 |
| 2010 AMC 10A | Triangle with AB=2AC, equilateral △CFE, angle condition; find ∠ACB | Geometry | Triangles & Polygons | 14 |
| 2010 AMC 10A | Four amphibians (toads always true, frogs always false) make statements; find number of frogs | Algebra | Logic & Truth Values | 15 |
| 2010 AMC 10A | Triangle ABC, angle bisector BD, AD=3, DC=8, integer sides; smallest perimeter | Geometry | Triangles & Polygons | 16 |
| 2010 AMC 10A | 3x3x3 cube with 2x2 square holes through each face; find remaining volume | Geometry | 3D Geometry & Solids | 17 |
| 2010 AMC 10A | Bernardo and Silvia pick three distinct digits and form descending numbers; probability Bernardo's is larger | Combinatorics | Probability | 18 |
| 2010 AMC 10A | Equiangular hexagon with side lengths 1 and r; area of △ACE is 70% of hexagon; sum of possible r | Geometry | Triangles & Polygons | 19 |
| 2010 AMC 10A | Fly visits all vertices of a unit cube, starting and ending same, each once; maximize path length | Geometry | 3D Geometry & Solids | 20 |
| 2010 AMC 10A | Cubic x³ - a x² + b x - 2010 with three positive integer roots; smallest possible a | Algebra | Polynomials | 21 |
| 2010 AMC 10A | Eight points on a circle, all chords drawn; count interior triangles with vertices inside circle | Combinatorics | Counting | 22 |
| 2010 AMC 10A | Boxes with red and white marbles; draw until red; find n where probability of stopping after n draws < 1/2010 | Combinatorics | Probability | 23 |
| 2010 AMC 10A | Last two non-zero digits of 90! | Number Theory | Modular Arithmetic | 24 |
| 2010 AMC 10A | Jim subtracts largest possible square repeatedly to reach zero; smallest n giving sequence length 8 | Algebra | Sequences & Series | 25 |
| 2010 AMC 10B | Evaluate the arithmetic expression 100(100-3) - (100*100 - 3). | Algebra | Systems & Algebraic Manipulation | 1 |
| 2010 AMC 10B | Find the percent of a 9-hour work day spent in two meetings totaling 135 minutes. | Algebra | Rates, Mixtures, & Averages | 2 |
| 2010 AMC 10B | Minimum socks to pull to guarantee a matching pair with at least 2 of each of 4 colors. | Combinatorics | Counting | 3 |
| 2010 AMC 10B | Define a custom average operation ♡(x); find the sum of values at 1, 2, and 3. | Algebra | Functions | 4 |
| 2010 AMC 10B | A 31-day month has equal Mondays and Wednesdays; how many possible first days? | Number Theory | Modular Arithmetic | 5 |
| 2010 AMC 10B | Given a circle with diameter AB, point C, and ∠COB=50°, find ∠CAB. | Geometry | Circular Geometry | 6 |
| 2010 AMC 10B | Isosceles triangle sides 10,10,12; rectangle width 4 with same area; find its perimeter. | Geometry | Triangles & Polygons | 7 |
| 2010 AMC 10B | Ticket costs x dollars; groups pay $48 and $64. How many possible whole-number values for x? | Number Theory | Factors, Multiples & Divisibility | 8 |
| 2010 AMC 10B | Larry ignores parentheses but gets the correct result; find the substituted number e. | Algebra | Systems & Algebraic Manipulation | 9 |
| 2010 AMC 10B | Shelby drives 16 miles in 40 minutes at two speeds; find minutes driven in the rain. | Algebra | Rates, Mixtures, & Averages | 10 |
| 2010 AMC 10B | Compare coupon discounts; find difference between max and min prices where A saves at least as much. | Algebra | Inequalities | 11 |
| 2010 AMC 10B | Survey Yes/No change from 50% to 70%; max minus min possible different answers. | Algebra | Systems & Algebraic Manipulation | 12 |
| 2010 AMC 10B | Find the sum of all solutions to the absolute value equation x = |2x - |60-2x||. | Algebra | Systems & Algebraic Manipulation | 13 |
| 2010 AMC 10B | The average of 1 through 99 and x is 100x; find x. | Algebra | Rates, Mixtures, & Averages | 14 |
| 2010 AMC 10B | Multiple-choice scoring: 4 for correct, -1 for wrong. Jesse scores 99; max number correct. | Algebra | Systems & Algebraic Manipulation | 15 |
| 2010 AMC 10B | Square of side 1 and circle of radius √3/3 share center; find area inside circle but outside square. | Geometry | Circular Geometry | 16 |
| 2010 AMC 10B | Andrea's median score among all; teammates placed 37th and 64th; find number of schools. | Algebra | Systems & Algebraic Manipulation | 17 |
| 2010 AMC 10B | Random a,b,c from 1–2010; probability that abc+ab+a is divisible by 3. | Combinatorics | Probability | 18 |
| 2010 AMC 10B | Circle area 156π, equilateral triangle ABC chord, OA=4√3; find side length of triangle. | Geometry | Circular Geometry | 19 |
| 2010 AMC 10B | Two circles tangent to regular hexagon and lines; find ratio of their areas. | Geometry | Circular Geometry | 20 |
| 2010 AMC 10B | Pick a random 4-digit palindrome; find the probability it is divisible by 7. | Combinatorics | Probability | 21 |
| 2010 AMC 10B | Distribute 7 distinct candies into 3 bags, red and blue each receive at least one piece. | Combinatorics | Counting | 22 |
| 2010 AMC 10B | Fill a 3x3 grid with digits 1-9 so rows and columns increase; count arrangements. | Combinatorics | Counting | 23 |
| 2010 AMC 10B | Basketball quarter scores form increasing geometric and arithmetic sequences; first-half total points. | Algebra | Sequences & Series | 24 |
| 2010 AMC 10B | Polynomial with integer coefficients has P(odd)=a, P(even)=-a; smallest possible a>0. | Algebra | Polynomials | 25 |
| 2009 AMC 10A | Find minimum number of 12 oz cans needed to make 128 oz. | Algebra | Rates, Mixtures, & Averages | 1 |
| 2009 AMC 10A | Determine which total value cannot be made from four coins (pennies, nickels, dimes, quarters). | Combinatorics | Counting | 2 |
| 2009 AMC 10A | Simplify the nested fraction 1 + 1/(1 + 1/(1+1)). | Algebra | Systems & Algebraic Manipulation | 3 |
| 2009 AMC 10A | Find required average bike speed for a triathlon given swim and run times. | Algebra | Rates, Mixtures, & Averages | 4 |
| 2009 AMC 10A | Find the sum of the digits of 111,111,111 squared. | Algebra | Sequences & Series | 5 |
| 2009 AMC 10A | Find the fraction of the semicircle's area that is shaded, given an inscribed circle. | Geometry | Circular Geometry | 6 |
| 2009 AMC 10A | Find whole milk fat percentage given 2% fat is 40% less than whole milk. | Algebra | Rates, Mixtures, & Averages | 7 |
| 2009 AMC 10A | Find total cost for family with children, adults, seniors given senior ticket price. | Algebra | Rates, Mixtures, & Averages | 8 |
| 2009 AMC 10A | Find a in geometric sequence a, b, 2009 with integer ratio. | Number Theory | Factors, Multiples & Divisibility | 9 |
| 2009 AMC 10A | Find area of right triangle given segments on hypotenuse from altitude. | Geometry | Triangles & Polygons | 10 |
| 2009 AMC 10A | Find volume of cube given volume change when dimensions are shifted by ±1. | Algebra | Systems & Algebraic Manipulation | 11 |
| 2009 AMC 10A | Find integer diagonal BD in quadrilateral with given sides using triangle inequality. | Geometry | Triangles & Polygons | 12 |
| 2009 AMC 10A | Express 12^{mn} in terms of P=2^m and Q=3^n. | Algebra | Logarithms & Exponents | 13 |
| 2009 AMC 10A | Find ratio of longer to shorter side of rectangles forming a square pattern. | Geometry | Triangles & Polygons | 14 |
| 2009 AMC 10A | Find number of diamonds in the 20th figure of a growing diamond pattern. | Algebra | Sequences & Series | 15 |
| 2009 AMC 10A | Find sum of all possible values of |a-d| given absolute differences of a,b,c,d. | Algebra | Systems & Algebraic Manipulation | 16 |
| 2009 AMC 10A | Find length EF where EF is perpendicular to diagonal DB in rectangle ABCD. | Geometry | Triangles & Polygons | 17 |
| 2009 AMC 10A | Find percentage of non-swimmers who play soccer given overlapping sport percentages. | Algebra | Rates, Mixtures, & Averages | 18 |
| 2009 AMC 10A | Find number of possible integer radii r<100 for rolling circle to align. | Number Theory | Factors, Multiples & Divisibility | 19 |
| 2009 AMC 10A | Find total time until Lauren reaches Andrea after Andrea stops with flat tire. | Algebra | Rates, Mixtures, & Averages | 20 |
| 2009 AMC 10A | Ratio of total area of four small circles to area of large enclosing circle. | Geometry | Circular Geometry | 21 |
| 2009 AMC 10A | Find probability sum is 7 after randomly reassigning numbers on two dice. | Combinatorics | Probability | 22 |
| 2009 AMC 10A | Find AE in quadrilateral with equal areas of two triangles formed by diagonals. | Geometry | Triangles & Polygons | 23 |
| 2009 AMC 10A | Probability that three random vertices of a cube determine a plane cutting through interior. | Combinatorics | Probability | 24 |
| 2009 AMC 10A | Find maximum number of factors of 2 in 10...064 with k zeros. | Number Theory | Factors, Multiples & Divisibility | 25 |
| 2009 AMC 10B | Jane buys 50-cent muffin or 75-cent bagel for 5 days; total cost whole dollars. How many bagels? | Algebra | Rates, Mixtures, & Averages | 1 |
| 2009 AMC 10B | Simplify the complex fraction ((1/3)-(1/4))/((1/2)-(1/3)). | Algebra | Systems & Algebraic Manipulation | 2 |
| 2009 AMC 10B | Painter had paint for 30 rooms, lost 3 cans, now only 25 rooms. Cans used for 25 rooms? | Algebra | Rates, Mixtures, & Averages | 3 |
| 2009 AMC 10B | Rectangular yard with two congruent isosceles right triangle flower beds; find fraction of yard occupied. | Geometry | Triangles & Polygons | 4 |
| 2009 AMC 10B | Twenty percent less than 60 is one-third more than what number? | Algebra | Systems & Algebraic Manipulation | 5 |
| 2009 AMC 10B | Product of ages of Kiana and her two older twin brothers is 128; find sum of their ages. | Number Theory | Factors, Multiples & Divisibility | 6 |
| 2009 AMC 10B | Insert parentheses in 2×3 + 4×5 to get different values; how many distinct values? | Combinatorics | Counting | 7 |
| 2009 AMC 10B | Gasoline price rises 20%, falls 20%, rises 25%, falls x%, returns to original; find x to nearest integer. | Algebra | Rates, Mixtures, & Averages | 8 |
| 2009 AMC 10B | Segments BD and AE intersect at C; AB=BC=CD=CE; ∠A = 5/2 ∠B. Find ∠D. | Geometry | Triangles & Polygons | 9 |
| 2009 AMC 10B | Flagpole 5 m tall snaps x m above ground; top touches ground 1 m from base. Find x. | Geometry | Triangles & Polygons | 10 |
| 2009 AMC 10B | How many 7-digit palindromes using digits 2,2,3,3,5,5,5? | Combinatorics | Counting | 11 |
| 2009 AMC 10B | Points A,B,C,D on one line, E,F on parallel line; max distinct areas of triangles from 6 points? | Geometry | Triangles & Polygons | 12 |
| 2009 AMC 10B | Convex pentagon rolled along x-axis; which side touches x=2009? | Geometry | Coordinate Geometry | 13 |
| 2009 AMC 10B | Millet seeds: add quart daily, birds eat 25% of millet; when does millet exceed half? | Algebra | Sequences & Series | 14 |
| 2009 AMC 10B | Bucket 2/3 full weighs a, 1/2 full weighs b; find full weight in terms of a and b. | Algebra | Systems & Algebraic Manipulation | 15 |
| 2009 AMC 10B | Circle, equilateral triangle ABC with BA, BC tangent; circle meets BO at D; find BD/BO. | Geometry | Circular Geometry | 16 |
| 2009 AMC 10B | Five unit squares; line from (c,0) to (3,3) divides region into equal areas. Find c. | Geometry | Coordinate Geometry | 17 |
| 2009 AMC 10B | Rectangle 8×6, M midpoint of AC, E on AB with ME ⟂ AC; find area of triangle AME. | Geometry | Triangles & Polygons | 18 |
| 2009 AMC 10B | Clock mistakenly shows 9 instead of 1; what fraction of day does it show correct time? | Combinatorics | Counting | 19 |
| 2009 AMC 10B | Right triangle ABC with AB=1, BC=2; angle bisector from A meets BC at D. Find BD. | Geometry | Triangles & Polygons | 20 |
| 2009 AMC 10B | Remainder when 3^0+3^1+⋯+3^{2009} is divided by 8. | Number Theory | Modular Arithmetic | 21 |
| 2009 AMC 10B | Cube cake with edge 2, iced on sides and top, cut vertically; find c+s for piece B. | Geometry | 3D Geometry & Solids | 22 |
| 2009 AMC 10B | Runners on circular track, laps 90s and 80s; picture taken randomly 10-11 min. Probability both in picture? | Combinatorics | Probability | 23 |
| 2009 AMC 10B | Keystone arch of 9 congruent isosceles trapezoids; find larger interior angle of trapezoid. | Geometry | Triangles & Polygons | 24 |
| 2009 AMC 10B | Cube with random stripe on each face; probability of a continuous stripe encircling the cube? | Combinatorics | Probability | 25 |
| 2008 AMC 10A | Doughnut machine starts 8:30 AM, finishes 1/3 at 11:10; find completion time. | Algebra | Rates, Mixtures, & Averages | 1 |
| 2008 AMC 10A | Square inside rectangle, width:side=2:1, length:width=2:1; find percent of rectangle area inside square. | Geometry | Triangles & Polygons | 2 |
| 2008 AMC 10A | Function <n> is sum of proper divisors; evaluate <<<6>>>. | Number Theory | Factors, Multiples & Divisibility | 3 |
| 2008 AMC 10A | If 2/3 of 10 bananas = 8 oranges, find oranges equal to 1/2 of 5 bananas. | Algebra | Rates, Mixtures, & Averages | 4 |
| 2008 AMC 10A | Compute product 8/4 * 12/8 * ... * 2008/2004. | Algebra | Sequences & Series | 5 |
| 2008 AMC 10A | Find average speed for triathlon with equal-distance legs at 3, 20, 10 km/h. | Algebra | Rates, Mixtures, & Averages | 6 |
| 2008 AMC 10A | Simplify (3^2008^2 - 3^2006^2)/(3^2007^2 - 3^2005^2). | Algebra | Logarithms & Exponents | 7 |
| 2008 AMC 10A | Heather compares 15% off + $90 rebate vs 25% off, saves $15; find sticker price. | Algebra | Systems & Algebraic Manipulation | 8 |
| 2008 AMC 10A | For (2x/3 - x/6) integer, which statement must be true about x? | Number Theory | Factors, Multiples & Divisibility | 9 |
| 2008 AMC 10A | Square S1 area 16, bisect sides to form smaller squares S2, S3; find S3 area. | Geometry | Triangles & Polygons | 10 |
| 2008 AMC 10A | Boat 1 mile from shore, water enters 10 gal/min, sinks >30 gal; row 4 mph; find slowest bailing rate. | Algebra | Rates, Mixtures, & Averages | 11 |
| 2008 AMC 10A | Red, blue, green marbles with percentage increases; express total in terms of r reds. | Algebra | Rates, Mixtures, & Averages | 12 |
| 2008 AMC 10A | Doug paints room in 5h, Dave in 7h, they work together with 1h break; find equation for total time t. | Algebra | Rates, Mixtures, & Averages | 13 |
| 2008 AMC 10A | Old TV 4:3 ratio, 27-inch diagonal; movie 2:1 ratio letterboxed; find height of each darkened strip. | Geometry | Triangles & Polygons | 14 |
| 2008 AMC 10A | Han drove 1h longer, 5mph faster than Ian, extra 70 mi; Jan 2h longer, 10mph faster; find Jan's extra miles. | Algebra | Rates, Mixtures, & Averages | 15 |
| 2008 AMC 10A | Two circles, one internally tangent and tangent to 60° angle rays; find ratio of areas. | Geometry | Circular Geometry | 16 |
| 2008 AMC 10A | Equilateral triangle side 6; find area of region outside triangle but within 3 units of its points. | Geometry | Triangles & Polygons | 17 |
| 2008 AMC 10A | Right triangle perimeter 32 and area 20; find the hypotenuse length. | Geometry | Triangles & Polygons | 18 |
| 2008 AMC 10A | Rectangle 2x6 rotated 90° twice; find length of path traveled by designated point. | Geometry | Circular Geometry | 19 |
| 2008 AMC 10A | Trapezoid bases 9 and 12, diagonals intersect at K; area of triangle AKD = 24; find trapezoid area. | Geometry | Triangles & Polygons | 20 |
| 2008 AMC 10A | Cube side 1; plane through two opposite vertices and midpoints of opposite edges; find quadrilateral area. | Geometry | 3D Geometry & Solids | 21 |
| 2008 AMC 10A | Sequence defined by coin flips (double/halve subtract 1); probability fourth term is integer. | Combinatorics | Probability | 22 |
| 2008 AMC 10A | Choose two subsets of {a,b,c,d,e} with union S and intersection exactly two; count ways (order irrelevant). | Combinatorics | Counting | 23 |
| 2008 AMC 10A | Let k=2008^2+2^2008; find the units digit of k^2+2^k. | Number Theory | Modular Arithmetic | 24 |
| 2008 AMC 10A | Six rectangular place mats (width 1, length x) around round table radius 4; find x. | Geometry | Circular Geometry | 25 |
| 2008 AMC 10B | Find number of possible total points from 5 baskets worth 2 or 3 points. | Combinatorics | Counting | 1 |
| 2008 AMC 10B | Calculate diagonal sum difference after row reversals in a 4x4 calendar. | Algebra | Systems & Algebraic Manipulation | 2 |
| 2008 AMC 10B | Simplify cube root of x times square root of x. | Algebra | Logarithms & Exponents | 3 |
| 2008 AMC 10B | Maximize a single player's salary given team total cap and minimum salary. | Algebra | Rates, Mixtures, & Averages | 4 |
| 2008 AMC 10B | Evaluate (x-y)^2 * (y-x)^2 under custom operation a*b=(a-b)^2. | Algebra | Functions | 5 |
| 2008 AMC 10B | Find what fraction BC is of AD given AB=4BD and AC=9CD on collinear points. | Algebra | Systems & Algebraic Manipulation | 6 |
| 2008 AMC 10B | How many unit equilateral triangles fill a large equilateral triangle of side 10? | Geometry | Triangles & Polygons | 7 |
| 2008 AMC 10B | Count the number of bouquets with roses and carnations costing exactly $50. | Algebra | Systems & Algebraic Manipulation | 8 |
| 2008 AMC 10B | Find the average of two real solutions to ax^2 - 2ax + b = 0. | Algebra | Polynomials | 9 |
| 2008 AMC 10B | Find length AC where C is the midpoint of minor arc AB on a circle. | Geometry | Circular Geometry | 10 |
| 2008 AMC 10B | Find u5 in a sequence satisfying u_{n+2}=2u_{n+1}+u_n given u3 and u6. | Algebra | Sequences & Series | 11 |
| 2008 AMC 10B | Convert pedometer readings and flips into miles walked by Postman Pete. | Algebra | Rates, Mixtures, & Averages | 12 |
| 2008 AMC 10B | Find the 2008th term of a sequence whose mean of first n terms is n. | Algebra | Sequences & Series | 13 |
| 2008 AMC 10B | Rotate point A in a 30-60-90 triangle 90° counterclockwise about origin and find coordinates. | Geometry | Coordinate Geometry | 14 |
| 2008 AMC 10B | Count integer right triangles with legs a,b, hypotenuse b+1, and b<100. | Number Theory | Factors, Multiples & Divisibility | 15 |
| 2008 AMC 10B | Probability sum of rolled dice is odd when coins determine number of dice rolled. | Combinatorics | Probability | 16 |
| 2008 AMC 10B | Probability exactly one of three randomly selected voters approves of mayor given 70% approval. | Combinatorics | Probability | 17 |
| 2008 AMC 10B | Find total bricks in a chimney given individual and combined work rates and output decrease. | Algebra | Rates, Mixtures, & Averages | 18 |
| 2008 AMC 10B | Volume of water in a horizontal cylindrical tank with given radius and water depth. | Geometry | 3D Geometry & Solids | 19 |
| 2008 AMC 10B | Probability sum of two nonstandard dice is 5, 7, or 9. | Combinatorics | Probability | 20 |
| 2008 AMC 10B | Count seating arrangements of 5 married couples alternating gender with spouse restrictions. | Combinatorics | Counting | 21 |
| 2008 AMC 10B | Probability that 3 red, 2 white, 1 blue beads in a line have no adjacent same color. | Combinatorics | Probability | 22 |
| 2008 AMC 10B | How many integer dimensions a<b for a rectangle with 1-foot border covering half area? | Algebra | Systems & Algebraic Manipulation | 23 |
| 2008 AMC 10B | Find angle BAD in quadrilateral with AB=BC=CD, angles 70° and 170°. | Geometry | Triangles & Polygons | 24 |
| 2008 AMC 10B | How many times do Michael and a garbage truck meet on a straight path with stops? | Algebra | Rates, Mixtures, & Averages | 25 |
| 2007 AMC 10A | Susan buys 4 tickets with 25% discount, Pam buys 5 with 30% discount; how much more does Pam pay? | Algebra | Rates, Mixtures, & Averages | 1 |
| 2007 AMC 10A | Custom operations a@b and a#b defined; evaluate (6@2)/(6#2). | Algebra | Functions | 2 |
| 2007 AMC 10A | Aquarium partially filled; brick placed inside; find water rise in cm. | Geometry | 3D Geometry & Solids | 3 |
| 2007 AMC 10A | Larger of two consecutive odd integers is 3 times smaller; find sum. | Algebra | Systems & Algebraic Manipulation | 4 |
| 2007 AMC 10A | Pencils and notebooks costs given; find total cost of 16 pencils and 10 notebooks. | Algebra | Systems & Algebraic Manipulation | 5 |
| 2007 AMC 10A | AMC 10 participation numbers 2002-2007; find largest percentage increase between consecutive years. | Algebra | Rates, Mixtures, & Averages | 6 |
| 2007 AMC 10A | Inheritance taxed 20% then 10% on remainder with total tax $10500; find inheritance. | Algebra | Rates, Mixtures, & Averages | 7 |
| 2007 AMC 10A | Isosceles triangles ABC and ADC share side AC; given two angles, find angle BAD. | Geometry | Triangles & Polygons | 8 |
| 2007 AMC 10A | Real numbers a,b satisfy 3^a = 81^{b+2} and 125^b = 5^{a-3}; find ab. | Algebra | Logarithms & Exponents | 9 |
| 2007 AMC 10A | Family average age 20, father 48, mother+children average 16; find number of children. | Algebra | Rates, Mixtures, & Averages | 10 |
| 2007 AMC 10A | Numbers 1-8 placed on cube vertices, face sums equal; find common sum. | Algebra | Systems & Algebraic Manipulation | 11 |
| 2007 AMC 10A | Two tour guides each must lead at least one of six tourists; number of possible groupings. | Combinatorics | Counting | 12 |
| 2007 AMC 10A | Carville to Nikpath takes 4.5 hr at 70 mph; find travel time at 60 mph. | Algebra | Rates, Mixtures, & Averages | 13 |
| 2007 AMC 10A | 3-4-5 triangle inscribed in circle of radius 3; find area. | Geometry | Triangles & Polygons | 14 |
| 2007 AMC 10A | Four small circles tangent to square and a central larger circle; find square area. | Geometry | Circular Geometry | 15 |
| 2007 AMC 10A | Integers a,b,c,d from 0 to 2007; probability ad-bc is even. | Combinatorics | Probability | 16 |
| 2007 AMC 10A | Positive integers m,n with 75m = n^3; minimize m+n. | Number Theory | Factors, Multiples & Divisibility | 17 |
| 2007 AMC 10A | 12-sided orthogonal polygon ABCDEFGHIJKL; find area of quadrilateral ABCM. | Geometry | Triangles & Polygons | 18 |
| 2007 AMC 10A | Square painted by sweeping brush along both diagonals; half area painted; find side to width ratio. | Geometry | Triangles & Polygons | 19 |
| 2007 AMC 10A | Given a + a^{-1} = 4, find a^4 + a^{-4}. | Algebra | Systems & Algebraic Manipulation | 20 |
| 2007 AMC 10A | Outer cube surface area 24, sphere inscribed, then inner cube inscribed in sphere; find inner cube surface area. | Geometry | 3D Geometry & Solids | 21 |
| 2007 AMC 10A | Sequence of three-digit numbers where tens/units become hundreds/tens of next term; largest prime dividing sum S. | Number Theory | Factors, Multiples & Divisibility | 22 |
| 2007 AMC 10A | Positive integers m>=n with m^2 - n^2 = 96; count ordered pairs. | Number Theory | Factors, Multiples & Divisibility | 23 |
| 2007 AMC 10A | Two circles radius 2 centered at A,B; tangents from midpoint O and common tangent EF form shaded region; find its area. | Geometry | Circular Geometry | 24 |
| 2007 AMC 10A | Count positive integers n with n + S(n) + S(S(n)) = 2007 where S(n) is digit sum. | Number Theory | Modular Arithmetic | 25 |
| 2007 AMC 10B | Isabella paints walls of 3 bedrooms each 12x10x8 feet with 60 sq ft openings; find total painted area. | Geometry | 3D Geometry & Solids | 1 |
| 2007 AMC 10B | Evaluate (3★5)-(5★3) where a★b=(a+b)b. | Algebra | Functions | 2 |
| 2007 AMC 10B | Student drives 120 mi at 30 mpg then returns at 20 mpg; find overall average gas mileage. | Algebra | Rates, Mixtures, & Averages | 3 |
| 2007 AMC 10B | Given central angles BOC=120°, AOB=140° in circumcircle of triangle ABC; find angle ABC. | Geometry | Circular Geometry | 4 |
| 2007 AMC 10B | Given 'all Arogs are Brafs, all Crups are Brafs, all Dramps are Arogs, all Crups are Dramps'; which statement is implied? | Algebra | Logic & Truth Values | 5 |
| 2007 AMC 10B | AMC 10 scoring: correct 6 pts, unanswered 1.5 pts. Sarah attempts 22, leaves 3 unanswered; how many correct to score at least 100? | Algebra | Inequalities | 6 |
| 2007 AMC 10B | Convex pentagon with equal sides, angles A and B each 90°; find angle E. | Geometry | Triangles & Polygons | 7 |
| 2007 AMC 10B | Digits of form bbcac with 0≤a<b<c≤9 and b average of a and c; how many such five-digit numbers? | Combinatorics | Counting | 8 |
| 2007 AMC 10B | Code shifts each occurrence of a letter right by triangular number; find letter that replaces 12th 's'. | Algebra | Sequences & Series | 9 |
| 2007 AMC 10B | Points B, C fixed; set S of points A such that triangle ABC has area 1; describe S. | Geometry | Triangles & Polygons | 10 |
| 2007 AMC 10B | Circle through vertices of isosceles triangle sides 3,3, base 2; find area of circle. | Geometry | Circular Geometry | 11 |
| 2007 AMC 10B | Tom's age T equals sum of his three children's ages; N years ago his age was twice sum of children's ages; find T/N. | Algebra | Systems & Algebraic Manipulation | 12 |
| 2007 AMC 10B | Two circles radius 2 centered at (2,0) and (0,2); find area of intersection. | Geometry | Circular Geometry | 13 |
| 2007 AMC 10B | Group initially 40% girls; 2 girls leave, 2 boys arrive, then 30% girls; find initial number of girls. | Algebra | Rates, Mixtures, & Averages | 14 |
| 2007 AMC 10B | Quadrilateral angles satisfy A=2B=3C=4D; find angle A rounded to nearest degree. | Algebra | Systems & Algebraic Manipulation | 15 |
| 2007 AMC 10B | Class 10% juniors, 90% seniors; overall average 84, senior average 83; find junior score. | Algebra | Rates, Mixtures, & Averages | 16 |
| 2007 AMC 10B | Point P inside equilateral triangle ABC; distances from P to sides 1,2,3; find side length AB. | Geometry | Triangles & Polygons | 17 |
| 2007 AMC 10B | Circle radius 1 surrounded by 4 equal circles of radius r; find r as shown. | Geometry | Circular Geometry | 18 |
| 2007 AMC 10B | Wheel spun twice, remainders mod 4 and 5 designate checkerboard square; find probability shaded square. | Combinatorics | Probability | 19 |
| 2007 AMC 10B | 5x5 grid of blocks; number of ways to choose 3 blocks with no two same row or column. | Combinatorics | Counting | 20 |
| 2007 AMC 10B | Square inscribed in right triangle 3-4-5 with two vertices on hypotenuse; find side length. | Geometry | Triangles & Polygons | 21 |
| 2007 AMC 10B | Choose 1-4 then roll two four-sided dice; win $1 if one shows number, $2 if both, lose $1 if none; find expected return. | Combinatorics | Probability | 22 |
| 2007 AMC 10B | Pyramid with square base cut by plane 2 units above base; smaller top pyramid surface area half original; find altitude. | Geometry | 3D Geometry & Solids | 23 |
| 2007 AMC 10B | Smallest positive integer divisible by 4 and 9 using only digits 4 and 9 (at least one each); find its last four digits. | Number Theory | Modular Arithmetic | 24 |
| 2007 AMC 10B | Number of coprime positive integer pairs (a,b) such that a/b + 14b/(9a) is integer. | Number Theory | Factors, Multiples & Divisibility | 25 |
| 2006 AMC 10A | Cost to purchase 5 sandwiches at $3 each and 8 sodas at $2 each. | Algebra | Rates, Mixtures, & Averages | 1 |
| 2006 AMC 10A | Evaluate h⊗(h⊗h) given the operation x⊗y = x³ - y. | Algebra | Functions | 2 |
| 2006 AMC 10A | Mary's age to Alice's age ratio 3:5; Alice is 30, find Mary's age. | Algebra | Rates, Mixtures, & Averages | 3 |
| 2006 AMC 10A | Find the largest possible sum of digits on a digital watch (hours, minutes, AM/PM). | Algebra | Systems & Algebraic Manipulation | 4 |
| 2006 AMC 10A | Pizza with 8 slices, plain $8, anchovy half $2 extra. Dave ate anchovy slices+1 plain. Find extra Dave paid. | Algebra | Rates, Mixtures, & Averages | 5 |
| 2006 AMC 10A | Solve (7x)¹⁴ = (14x)⁷ for non-zero real x. | Algebra | Logarithms & Exponents | 6 |
| 2006 AMC 10A | Rectangle 8×18 cut into two congruent hexagons; rearranged to square. Find segment y. | Geometry | Triangles & Polygons | 7 |
| 2006 AMC 10A | Find c for parabola y=x²+bx+c passing through (2,3) and (4,3). | Algebra | Systems & Algebraic Manipulation | 8 |
| 2006 AMC 10A | How many sets of ≥2 consecutive positive integers sum to 15? | Algebra | Sequences & Series | 9 |
| 2006 AMC 10A | How many real x make √(120-√x) an integer? | Algebra | Systems & Algebraic Manipulation | 10 |
| 2006 AMC 10A | Describe the graph of (x+y)² = x² + y². | Algebra | Systems & Algebraic Manipulation | 11 |
| 2006 AMC 10A | Dog tied with 8-ft rope to a 16-ft square shed; compare roaming areas of two arrangements. | Geometry | Circular Geometry | 12 |
| 2006 AMC 10A | Fair game: pay $5; roll die twice: win if both even and match. Find winning prize. | Combinatorics | Probability | 13 |
| 2006 AMC 10A | Rings of thickness 1 cm, diameters decreasing from 20 to 3 cm. Find total height. | Algebra | Sequences & Series | 14 |
| 2006 AMC 10A | Two runners on concentric circular tracks, opposite directions, different radii and speeds. Number of meetings in 30 min. | Algebra | Rates, Mixtures, & Averages | 15 |
| 2006 AMC 10A | Two tangent circles of radii 1 and 2, isosceles triangle tangent to both circles. Find triangle area. | Geometry | Triangles & Polygons | 16 |
| 2006 AMC 10A | In rectangle, trisected sides, find area of central quadrilateral formed by diagonals. | Geometry | Triangles & Polygons | 17 |
| 2006 AMC 10A | Count distinct license plates with 4 digits and 2 adjacent letters (any order, repeating allowed). | Combinatorics | Counting | 18 |
| 2006 AMC 10A | Count non-similar triangles with angles as distinct positive integers in arithmetic progression. | Algebra | Sequences & Series | 19 |
| 2006 AMC 10A | Probability that among six distinct integers from 1-2006, some pair's difference is a multiple of 5. | Combinatorics | Probability | 20 |
| 2006 AMC 10A | Count 4-digit integers containing at least one digit 2 or 3. | Combinatorics | Counting | 21 |
| 2006 AMC 10A | Pigs $300, goats $210; find smallest positive debt expressible as 300p+210g. | Number Theory | Factors, Multiples & Divisibility | 22 |
| 2006 AMC 10A | Two circles radii 3 and 8 with internal tangent CD, AB meets CD at E, AE=5. Find CD. | Geometry | Circular Geometry | 23 |
| 2006 AMC 10A | Find volume of regular octahedron formed by joining centers of adjacent faces of unit cube. | Geometry | 3D Geometry & Solids | 24 |
| 2006 AMC 10A | Prob bug visits all 8 vertices exactly once in 7 random edge moves on a cube. | Combinatorics | Probability | 25 |
| 2006 AMC 10B | Sum of alternating powers of -1 from 1 to 2006 | Algebra | Sequences & Series | 1 |
| 2006 AMC 10B | Custom operation (x+y)(x-y); compute 3 ♠ (4 ♠ 5) | Algebra | Functions | 2 |
| 2006 AMC 10B | Total points 34, win margin 14; find Panthers' score | Algebra | Systems & Algebraic Manipulation | 3 |
| 2006 AMC 10B | Concentric circles diameter 1 and 3; ratio of ring area to inner circle | Geometry | Circular Geometry | 4 |
| 2006 AMC 10B | Pack 2x3 and 3x4 rectangles in smallest square without overlap | Geometry | Triangles & Polygons | 5 |
| 2006 AMC 10B | Semicircular arcs on square side 2/π; find perimeter of region | Geometry | Circular Geometry | 6 |
| 2006 AMC 10B | Simplify sqrt(x/(1-(x-1)/x)) for x<0 | Algebra | Systems & Algebraic Manipulation | 7 |
| 2006 AMC 10B | Square area 40 inscribed in semicircle; find area of semicircle | Geometry | Circular Geometry | 8 |
| 2006 AMC 10B | Lemonade with 100g lemon, 100g sugar, 400g water; calories in 200g | Algebra | Rates, Mixtures, & Averages | 9 |
| 2006 AMC 10B | Triangle sides: one 3x another, third 15; greatest integer perimeter | Geometry | Triangles & Polygons | 10 |
| 2006 AMC 10B | Tens digit of sum 7!+8!+...+2006! | Number Theory | Modular Arithmetic | 11 |
| 2006 AMC 10B | Lines x=y/4+a, y=x/4+b intersect at (1,2); find a+b | Algebra | Systems & Algebraic Manipulation | 12 |
| 2006 AMC 10B | Joe drinks 2 oz then adds cream; JoAnn adds cream then drinks 2 oz; cream ratio | Algebra | Rates, Mixtures, & Averages | 13 |
| 2006 AMC 10B | Roots a,b of x^2-mx+2=0; new roots a+1/b, b+1/a; find product q | Algebra | Polynomials | 14 |
| 2006 AMC 10B | Similar rhombi ABCD and BFDE, angle BAD=60°, area ABCD=24; find area BFDE | Geometry | Triangles & Polygons | 15 |
| 2006 AMC 10B | Feb 29, 2004 is Sunday; day of week for Feb 29, 2020 | Number Theory | Modular Arithmetic | 16 |
| 2006 AMC 10B | Bag swap: Alice moves one ball, Bob moves one; probability bags identical | Combinatorics | Probability | 17 |
| 2006 AMC 10B | Recursive sequence a_n=a_{n-1}/a_{n-2}, a1=2,a2=3; find a2006 | Algebra | Sequences & Series | 18 |
| 2006 AMC 10B | Circle radius 2, square OABC side 1; find shaded area bounded by BD, BE, arc DE | Geometry | Circular Geometry | 19 |
| 2006 AMC 10B | Rectangle ABCD with A(6,-22), B(2006,178), D(8,y); find area | Geometry | Coordinate Geometry | 20 |
| 2006 AMC 10B | Dice probabilities in ratio 1:2:3:4:5:6; probability of sum 7 | Combinatorics | Probability | 21 |
| 2006 AMC 10B | Sandwiches cost 4¢B+5¢J per, total $2.53; find cost of jam | Number Theory | Factors, Multiples & Divisibility | 22 |
| 2006 AMC 10B | Triangle partitioned into 3 triangles and quadrilateral; areas 3,7,7; find shaded area | Geometry | Triangles & Polygons | 23 |
| 2006 AMC 10B | Externally tangent circles radii 2 and 4; common tangents; area of hexagon AOBCPD | Geometry | Circular Geometry | 24 |
| 2006 AMC 10B | 8 children ages include 9; 4-digit number divisible by all; which age not possible? | Number Theory | Factors, Multiples & Divisibility | 25 |
| 2005 AMC 10A | Mike and Joe each tip $2, at 10% and 20% of their bills. Find bill difference. | Algebra | Rates, Mixtures, & Averages | 1 |
| 2005 AMC 10A | Custom operation (a★b)=(a+b)/(a-b). Compute ((1★2)★3). | Algebra | Functions | 2 |
| 2005 AMC 10A | Equations 2x+7=3 and bx-10=-2 share same solution. Find b. | Algebra | Systems & Algebraic Manipulation | 3 |
| 2005 AMC 10A | Rectangle diagonal x, length twice width. Express area in terms of x. | Geometry | Triangles & Polygons | 4 |
| 2005 AMC 10A | Store offers one free per four purchased. Dave needs 7, Doug 8. Savings together vs. separately. | Algebra | Rates, Mixtures, & Averages | 5 |
| 2005 AMC 10A | Average of 20 numbers is 30, average of another 30 is 20. Find overall average of 50 numbers. | Algebra | Rates, Mixtures, & Averages | 6 |
| 2005 AMC 10A | Josh rides twice as long as Mike at four-fifths his speed. They meet after 13 miles total. Find Mike's distance. | Algebra | Rates, Mixtures, & Averages | 7 |
| 2005 AMC 10A | Square with side sqrt(50), inscribed inner square with BE=1. Find area of inner square. | Geometry | Triangles & Polygons | 8 |
| 2005 AMC 10A | Three X and two O tiles randomly arranged. Probability of reading XOXOX. | Combinatorics | Probability | 9 |
| 2005 AMC 10A | Quadratic 4x^2+ax+8x+9=0 has one solution. Find sum of possible a. | Algebra | Polynomials | 10 |
| 2005 AMC 10A | Cube painted red, cut into unit cubes; exactly 1/4 of all faces red. Find n. | Geometry | 3D Geometry & Solids | 11 |
| 2005 AMC 10A | Trefoil from circular sectors on equilateral triangles, base length 2. Find area. | Geometry | Circular Geometry | 12 |
| 2005 AMC 10A | Inequality (130n)^50 > n^100 > 2^200. Count positive integer solutions n. | Algebra | Inequalities | 13 |
| 2005 AMC 10A | Three-digit numbers where middle digit is average of first and last digits. How many? | Combinatorics | Counting | 14 |
| 2005 AMC 10A | How many positive cubes divide 3! * 5! * 7! exactly? | Number Theory | Factors, Multiples & Divisibility | 15 |
| 2005 AMC 10A | Two-digit number minus sum of digits yields unit digit 6. Count possibilities. | Algebra | Systems & Algebraic Manipulation | 16 |
| 2005 AMC 10A | Five-pointed star with vertices 3,5,6,7,9; edge sums form arithmetic progression. Find middle sum. | Algebra | Sequences & Series | 17 |
| 2005 AMC 10A | Best-of-5 series; A wins series, B wins game 2. Probability B won game 1. | Combinatorics | Probability | 18 |
| 2005 AMC 10A | Rotated unit square lowered between two fixed unit squares. Find distance of lifted vertex from base. | Geometry | Triangles & Polygons | 19 |
| 2005 AMC 10A | Equiangular octagon with alternating sides 1 and sqrt(2)/2. Find area. | Geometry | Triangles & Polygons | 20 |
| 2005 AMC 10A | Sum 1+2+...+n evenly divides 6n. Count positive integers n. | Number Theory | Factors, Multiples & Divisibility | 21 |
| 2005 AMC 10A | S is first 2005 multiples of 4, T first 2005 multiples of 6. How many common elements? | Number Theory | Factors, Multiples & Divisibility | 22 |
| 2005 AMC 10A | Diameter AB with point C (2*AC=BC), DC perp AB, DE diameter. Ratio of areas of triangles DCE and ABD. | Geometry | Circular Geometry | 23 |
| 2005 AMC 10A | Both P(n)=sqrt(n) and P(n+48)=sqrt(n+48) hold (P is greatest prime factor). How many n? | Number Theory | Factors, Multiples & Divisibility | 24 |
| 2005 AMC 10A | Triangle ABC with side lengths 25,39,42. Points D and E on AB, AC divide. Ratio of area ADE to quadrilateral BCED. | Geometry | Triangles & Polygons | 25 |
| 2005 AMC 10B | Scout troop buys 1000 candy bars at 5 for $2, sells at 2 for $1; find profit. | Algebra | Rates, Mixtures, & Averages | 1 |
| 2005 AMC 10B | Find positive number x such that x% of x equals 4. | Algebra | Systems & Algebraic Manipulation | 2 |
| 2005 AMC 10B | Gallon of paint; 1/3 used first day, 1/3 of remainder second; fraction left for third day. | Algebra | Rates, Mixtures, & Averages | 3 |
| 2005 AMC 10B | Evaluate (5⋄12)⋄((-12)⋄(-5)) where a⋄b = √(a²+b²). | Algebra | Functions | 4 |
| 2005 AMC 10B | Brianna spends 1/5 of money to buy 1/3 of CDs; what fraction of money remains after buying all? | Algebra | Rates, Mixtures, & Averages | 5 |
| 2005 AMC 10B | Lisa aims for A on 80% of 50 quizzes; has 22 A's after 30; how many more non-A's can she afford? | Algebra | Rates, Mixtures, & Averages | 6 |
| 2005 AMC 10B | Circle inscribed in square, then square inscribed in that circle, then circle inscribed; find area ratio smallest circle to largest square. | Geometry | Circular Geometry | 7 |
| 2005 AMC 10B | An 8×10 ft floor tiled with 1-ft tiles, each with four quarter-circles; find total shaded area. | Geometry | Circular Geometry | 8 |
| 2005 AMC 10B | Two dice with faces 1,1,2,2,3,3 and 4,4,5,5,6,6; find probability sum is odd. | Combinatorics | Probability | 9 |
| 2005 AMC 10B | Isosceles triangle legs 7, base 2; point D on extended base with CD=8; find BD. | Geometry | Triangles & Polygons | 10 |
| 2005 AMC 10B | Sequence starts 2005, each term is sum of cubes of digits; find 2005th term. | Algebra | Sequences & Series | 11 |
| 2005 AMC 10B | Twelve fair dice rolled; find probability product of numbers is prime. | Combinatorics | Probability | 12 |
| 2005 AMC 10B | Count numbers 1–2005 that are multiples of 3 or 4 but not 12. | Number Theory | Factors, Multiples & Divisibility | 13 |
| 2005 AMC 10B | Equilateral triangle side 2, M midpoint of AC, C midpoint of BD; find area of triangle CDM. | Geometry | Triangles & Polygons | 14 |
| 2005 AMC 10B | Eight bills: 2 each of $1, $5, $10, $20; draw two; find probability sum ≥ $20. | Combinatorics | Probability | 15 |
| 2005 AMC 10B | Quadratic x²+mx+n has roots twice those of x²+px+m; find n/p. | Algebra | Polynomials | 16 |
| 2005 AMC 10B | If 4^a=5, 5^b=6, 6^c=7, 7^d=8, find a·b·c·d. | Algebra | Logarithms & Exponents | 17 |
| 2005 AMC 10B | Phone numbers 555-abc-defg, digits a–g distinct, increasing, none 0 or 1; how many possible numbers? | Combinatorics | Counting | 18 |
| 2005 AMC 10B | Score distribution: 10% 70, 25% 80, 20% 85, 15% 90, rest 95; find mean minus median. | Algebra | Rates, Mixtures, & Averages | 19 |
| 2005 AMC 10B | Average of all 5-digit numbers using digits 1,3,5,7,8 exactly once each. | Combinatorics | Counting | 20 |
| 2005 AMC 10B | Forty slips, numbers 1–10 each four times; draw four; find ratio q/p where p = all same number, q = two of one and two of another. | Combinatorics | Probability | 21 |
| 2005 AMC 10B | For how many n ≤ 24 is n! divisible by 1+2+...+n? | Number Theory | Factors, Multiples & Divisibility | 22 |
| 2005 AMC 10B | Trapezoid ABCD, AB||DC, midpoints E,F of BC,DA; area ABEF twice area FECD; find AB/DC. | Geometry | Triangles & Polygons | 23 |
| 2005 AMC 10B | Two-digit x and reversed y satisfy x²−y²=m²; find x+y+m. | Number Theory | Factors, Multiples & Divisibility | 24 |
| 2005 AMC 10B | Maximum size of subset of {1,…,100} with no two elements summing to 125. | Combinatorics | Counting | 25 |
| 2004 AMC 10A | Six people split $1500 equally; find each person's amount. | Algebra | Rates, Mixtures, & Averages | 1 |
| 2004 AMC 10A | Given custom operation ⊗(a,b,c)=a/(b-c), evaluate a nested expression. | Algebra | Functions | 2 |
| 2004 AMC 10A | Alicia earns $20/hr with 1.45% tax; find tax in cents per hour. | Algebra | Rates, Mixtures, & Averages | 3 |
| 2004 AMC 10A | Solve |x-1| = |x-2| for x. | Algebra | Systems & Algebraic Manipulation | 4 |
| 2004 AMC 10A | Probability that three random points from a 3x3 grid are collinear. | Combinatorics | Probability | 5 |
| 2004 AMC 10A | Bertha's daughters and granddaughters total 30; find how many have no daughters. | Algebra | Rates, Mixtures, & Averages | 6 |
| 2004 AMC 10A | Oranges stacked in a pyramid with 5x8 base; each layer reduces; find total oranges. | Algebra | Sequences & Series | 7 |
| 2004 AMC 10A | Token game: players start 15,14,13; each round leader gives away tokens; how many rounds? | Algebra | Sequences & Series | 8 |
| 2004 AMC 10A | Overlapping right triangles; find difference of areas of two smaller triangles. | Geometry | Triangles & Polygons | 9 |
| 2004 AMC 10A | Probability that number of heads is equal when coin A flipped 3 times and B 4 times. | Combinatorics | Probability | 10 |
| 2004 AMC 10A | Jar diameter increased 25% while volume unchanged; find percent height decrease. | Geometry | 3D Geometry & Solids | 11 |
| 2004 AMC 10A | Hamburger: 3 meat choices and any subset of 8 condiments; total number of kinds. | Combinatorics | Counting | 12 |
| 2004 AMC 10A | 12 men each dance with 3 women, each woman with 2 men; find number of women. | Combinatorics | Counting | 13 |
| 2004 AMC 10A | Average coin value 20 cents; adding one quarter makes average 21; how many dimes? | Algebra | Rates, Mixtures, & Averages | 14 |
| 2004 AMC 10A | Given -4 ≤ x ≤ -2, 2 ≤ y ≤ 4, maximize (x+y)/x. | Algebra | Inequalities | 15 |
| 2004 AMC 10A | 5x5 grid of squares; count squares that contain the center black square. | Combinatorics | Counting | 16 |
| 2004 AMC 10A | Two runners start diametrically opposite on circular track; find track length from meeting points. | Algebra | Rates, Mixtures, & Averages | 17 |
| 2004 AMC 10A | Arithmetic progression 9, ...; modified to geometric progression; smallest third term? | Algebra | Sequences & Series | 18 |
| 2004 AMC 10A | Cylinder stripe width 3 makes two revolutions; find stripe area. | Geometry | 3D Geometry & Solids | 19 |
| 2004 AMC 10A | Square with equilateral triangle inside; ratio of areas of two triangles. | Geometry | Triangles & Polygons | 20 |
| 2004 AMC 10A | Two lines through center of three concentric circles; shaded area 8/13 of unshaded; find angle. | Geometry | Circular Geometry | 21 |
| 2004 AMC 10A | Square side 2, tangent from vertex to semicircle meets side; find segment length. | Geometry | Circular Geometry | 22 |
| 2004 AMC 10A | Three circles externally tangent, internally tangent to larger circle; find radius of smaller. | Geometry | Circular Geometry | 23 |
| 2004 AMC 10A | Function defined by f(1)=1, f(2n)=n·f(n); evaluate f(2^100). | Algebra | Functions | 24 |
| 2004 AMC 10A | Three unit spheres on plane support sphere radius 2; distance from plane to top. | Geometry | 3D Geometry & Solids | 25 |
| 2004 AMC 10B | Find the number of seats in rows 12 through 22 of an amphitheater with 33 seats per row. | Algebra | Rates, Mixtures, & Averages | 1 |
| 2004 AMC 10B | How many two-digit positive integers have at least one digit as 7? | Combinatorics | Counting | 2 |
| 2004 AMC 10B | Jenny doubles her free throws each practice; made 48 at fifth practice; find her first practice amount. | Algebra | Sequences & Series | 3 |
| 2004 AMC 10B | Roll a six-sided die; product P of the five visible numbers; find largest certain divisor of P. | Number Theory | Factors, Multiples & Divisibility | 4 |
| 2004 AMC 10B | Maximize c·a^b - d using the digits 0,1,2,3 each exactly once. | Algebra | Systems & Algebraic Manipulation | 5 |
| 2004 AMC 10B | Which product of two factorials among the choices is a perfect square? | Number Theory | Factors, Multiples & Divisibility | 6 |
| 2004 AMC 10B | Isabella exchanges US dollars for Canadian dollars at 7:10 rate, spends 60 CAD, ends with d CAD; find digit sum of d. | Algebra | Rates, Mixtures, & Averages | 7 |
| 2004 AMC 10B | Airport is 8 miles SW of St. Paul and 10 miles SE of Minneapolis; find the distance between the two downtowns. | Geometry | Triangles & Polygons | 8 |
| 2004 AMC 10B | Square side 10 and circle radius 10 centered at one vertex; find area of their union. | Geometry | Circular Geometry | 9 |
| 2004 AMC 10B | Display of cans: top row 1, each lower row has 2 more than row above; total 100 cans; how many rows? | Algebra | Sequences & Series | 10 |
| 2004 AMC 10B | Two eight-sided dice rolled; find probability that product of the two top numbers exceeds their sum. | Combinatorics | Probability | 11 |
| 2004 AMC 10B | Annulus between radii b and c, with tangent segment a; express its area in terms of a, b, c, d, or e. | Geometry | Circular Geometry | 12 |
| 2004 AMC 10B | Stack of US coins exactly 14 mm tall with given thicknesses; how many coins are in the stack? | Algebra | Systems & Algebraic Manipulation | 13 |
| 2004 AMC 10B | Bag of red and blue marbles; add red, then yellow, then double blue; find final fraction blue. | Algebra | Rates, Mixtures, & Averages | 14 |
| 2004 AMC 10B | Patty has 20 nickels and dimes; if swapped, value would be 70 cents more; find her coins' total worth. | Algebra | Systems & Algebraic Manipulation | 15 |
| 2004 AMC 10B | Three unit circles externally tangent to each other and internally tangent to a larger circle; find large radius. | Geometry | Circular Geometry | 16 |
| 2004 AMC 10B | Jack's age digits are Bill's reversed; in 5 years Jack will be twice Bill; find their current age difference. | Algebra | Systems & Algebraic Manipulation | 17 |
| 2004 AMC 10B | Right triangle ACE, points B, D, F on sides with given lengths; find area ratio of triangle DBF to ACE. | Geometry | Triangles & Polygons | 18 |
| 2004 AMC 10B | Sequence defined by a_n = a_{n-2}+a_{n-1}-a_{n-3}; find the 2004th term given first three terms 2001,2002,2003. | Algebra | Sequences & Series | 19 |
| 2004 AMC 10B | Triangle with cevians AD and BE intersecting at T; AT/DT=3, BT/ET=4; find CD/BD. | Geometry | Triangles & Polygons | 20 |
| 2004 AMC 10B | Union of first 2004 terms of two arithmetic progressions: 1,4,7,... and 9,16,23,...; how many distinct numbers? | Number Theory | Modular Arithmetic | 21 |
| 2004 AMC 10B | Right triangle with sides 5,12,13; find distance between the centers of its inscribed and circumscribed circles. | Geometry | Circular Geometry | 22 |
| 2004 AMC 10B | Each face of a cube painted red/blue with equal independent probability; probability that four vertical faces can be made all same color when placed on a surface. | Combinatorics | Probability | 23 |
| 2004 AMC 10B | Triangle ABC with sides 7,8,9; D on circumcircle so AD bisects angle A; find AD/CD. | Geometry | Circular Geometry | 24 |
| 2004 AMC 10B | Circle radius 1 internally tangent to two circles radius 2 at ends of its diameter; find shaded area outside small circle but inside the large circles. | Geometry | Circular Geometry | 25 |
| 2003 AMC 10A | Difference between sum of first 2003 even and sum of first 2003 odd counting numbers | Algebra | Sequences & Series | 1 |
| 2003 AMC 10A | Socks $4, T-shirt $5 more, each member needs 2 socks and 2 shirts, total $2366; find members | Algebra | Rates, Mixtures, & Averages | 2 |
| 2003 AMC 10A | 15x10x8 cm box, 3 cm cube removed from each corner; percent volume removed | Geometry | 3D Geometry & Solids | 3 |
| 2003 AMC 10A | Anna walks 1 km uphill 30 min, downhill 10 min; average speed for round trip | Algebra | Rates, Mixtures, & Averages | 4 |
| 2003 AMC 10A | Roots d,e of 2x^2+3x-5=0; find (d-1)(e-1) | Algebra | Polynomials | 5 |
| 2003 AMC 10A | Define x ♡ y = |x-y|; which statement is not true? | Algebra | Functions | 6 |
| 2003 AMC 10A | How many noncongruent triangles with perimeter 7 have integer side lengths? | Combinatorics | Counting | 7 |
| 2003 AMC 10A | Probability randomly chosen positive factor of 60 is less than 7 | Combinatorics | Probability | 8 |
| 2003 AMC 10A | Simplify nested radicals: ∛(x∛(x∛(x√x))) | Algebra | Logarithms & Exponents | 9 |
| 2003 AMC 10A | Attach one square to 4-square polygon at 9 positions; how many fold to cube missing one face? | Geometry | 3D Geometry & Solids | 10 |
| 2003 AMC 10A | AMC10 + AMC12 = 123422; find A+M+C | Algebra | Systems & Algebraic Manipulation | 11 |
| 2003 AMC 10A | Random point in rectangle (0,0)-(4,1); probability x<y | Combinatorics | Probability | 12 |
| 2003 AMC 10A | Three numbers sum to 20, first 4× sum of others, second 7× third; find product | Algebra | Systems & Algebraic Manipulation | 13 |
| 2003 AMC 10A | Largest integer product of distinct primes d, e, 10d+e with single-digit d,e; sum of its digits | Number Theory | Factors, Multiples & Divisibility | 14 |
| 2003 AMC 10A | Probability integer 1..100 divisible by 2 and not by 3 | Combinatorics | Probability | 15 |
| 2003 AMC 10A | Units digit of 13^2003 | Number Theory | Modular Arithmetic | 16 |
| 2003 AMC 10A | Perimeter of equilateral triangle equals area of its circumscribed circle; find radius | Geometry | Circular Geometry | 17 |
| 2003 AMC 10A | Sum of reciprocals of roots of (2003/2004)x+1+1/x=0 | Algebra | Polynomials | 18 |
| 2003 AMC 10A | Area of lune between semicircle diameter 1 atop semicircle diameter 2 | Geometry | Circular Geometry | 19 |
| 2003 AMC 10A | Probability base-10 3-digit number also 3-digit in base 9 and base 11 | Combinatorics | Probability | 20 |
| 2003 AMC 10A | Select 6 cookies from 3 types, at least 6 of each; how many assortments? | Combinatorics | Counting | 21 |
| 2003 AMC 10A | Rectangle ABCD, BH=6, DE=4, find GF where lines intersect | Geometry | Triangles & Polygons | 22 |
| 2003 AMC 10A | Toothpick equilateral triangle with base row of 2003 small triangles; total toothpicks | Algebra | Sequences & Series | 23 |
| 2003 AMC 10A | Stack 5 red 1-5 and 4 blue 3-6 alternating, red divides neighboring blue; sum middle three | Combinatorics | Counting | 24 |
| 2003 AMC 10A | n is 5-digit, q quotient, r remainder when divided by 100; how many n with q+r divisible by 11? | Number Theory | Modular Arithmetic | 25 |
| 2003 AMC 10B | Simplify the fraction (2-4+6-8+10-12+14)/(3-6+9-12+15-18+21). | Algebra | Systems & Algebraic Manipulation | 1 |
| 2003 AMC 10B | Al takes one green and one pink pill daily for two weeks; total cost $546, green costs $1 more. | Algebra | Rates, Mixtures, & Averages | 2 |
| 2003 AMC 10B | Sum of 5 consecutive even integers is 4 less than sum of first 8 odd numbers. Find smallest even. | Algebra | Sequences & Series | 3 |
| 2003 AMC 10B | Rectangular flower bed with five regions; minimize planting cost by assigning flower types to areas. | Algebra | Rates, Mixtures, & Averages | 4 |
| 2003 AMC 10B | Moe mows a 90x150 ft lawn with 28-inch swath overlapping 4 inches, speed 5000 ft/hr. Time? | Algebra | Rates, Mixtures, & Averages | 5 |
| 2003 AMC 10B | TV screen diagonal 27 inches, aspect ratio 4:3. Find horizontal length. | Geometry | Triangles & Polygons | 6 |
| 2003 AMC 10B | Compute sum of floor(sqrt(n)) for n from 1 to 16. | Algebra | Sequences & Series | 7 |
| 2003 AMC 10B | Geometric sequence: second term 2, fourth term 6. Find possible first term. | Algebra | Sequences & Series | 8 |
| 2003 AMC 10B | Solve 25^{-2} = 5^{48/x} / (5^{26/x} * 25^{17/x}) for x. | Algebra | Logarithms & Exponents | 9 |
| 2003 AMC 10B | Old license plates: 1 letter + 4 digits. New: 3 letters + 3 digits. Find increase factor. | Combinatorics | Counting | 10 |
| 2003 AMC 10B | Lines with slopes 3 and 5 intersect at (10,15). Find distance between their x-intercepts. | Geometry | Coordinate Geometry | 11 |
| 2003 AMC 10B | Al, Betty, Clare split $1000. After one year, Al loses $100, others double. Total $1500. Find Al's original. | Algebra | Systems & Algebraic Manipulation | 12 |
| 2003 AMC 10B | For two-digit x, digit sum function ♣(x). Find number of x with ♣(♣(x))=3. | Number Theory | Factors, Multiples & Divisibility | 13 |
| 2003 AMC 10B | Express 3^8 * 5^2 as a^b with positive integers a,b; minimize a+b. | Number Theory | Factors, Multiples & Divisibility | 14 |
| 2003 AMC 10B | 100-player single elimination tennis tournament, 28 byes. Total matches played? | Combinatorics | Counting | 15 |
| 2003 AMC 10B | Restaurant: 3 desserts, appetizers = 2 * main courses. Dinners must cover 365 days. Minimum main courses? | Algebra | Inequalities | 16 |
| 2003 AMC 10B | Ice cream cone: sphere of ice cream melts to fill cone at 75% volume. Ratio height:radius? | Geometry | 3D Geometry & Solids | 17 |
| 2003 AMC 10B | Largest integer dividing (n+1)(n+3)(n+5)(n+7)(n+9) for all positive even n. | Number Theory | Factors, Multiples & Divisibility | 18 |
| 2003 AMC 10B | Three unit semicircles on diameter of larger semicircle radius 2. Shaded area within large semicircle. | Geometry | Circular Geometry | 19 |
| 2003 AMC 10B | Rectangle ABCD, points F,G on CD, lines AF and BG intersect at E. Find area of triangle AEB. | Geometry | Triangles & Polygons | 20 |
| 2003 AMC 10B | Bag: 2 red, 2 green beads. Draw and replace with red each time. Probability all red after 3 replacements. | Combinatorics | Probability | 21 |
| 2003 AMC 10B | Clock chimes once at half-hour, hour number on hour. Starting 11:15 AM Feb 26, 2003. Date of 2003rd chime? | Algebra | Sequences & Series | 22 |
| 2003 AMC 10B | Regular octagon area 1. Find area of rectangle formed by two opposite sides. | Geometry | Triangles & Polygons | 23 |
| 2003 AMC 10B | Arithmetic sequence: first four terms are x+y, x-y, xy, x/y. Find fifth term. | Algebra | Sequences & Series | 24 |
| 2003 AMC 10B | How many 4-digit numbers divisible by 3 and ending in 23? | Number Theory | Modular Arithmetic | 25 |
| 2002 AMC 10A | Find the value closest to a fraction involving powers of 10. | Algebra | Logarithms & Exponents | 1 |
| 2002 AMC 10A | Evaluate a custom-defined three-variable operation for given numbers. | Algebra | Functions | 2 |
| 2002 AMC 10A | Count distinct values from parenthesizing 2^2^2^2 differently. | Algebra | Logarithms & Exponents | 3 |
| 2002 AMC 10A | Find how many positive integers m satisfy m*n ≤ m+n for some n. | Algebra | Inequalities | 4 |
| 2002 AMC 10A | Area of shaded region between concentric circles and smaller tangent circles. | Geometry | Circular Geometry | 5 |
| 2002 AMC 10A | Correct an arithmetic mistake by working backwards. | Algebra | Systems & Algebraic Manipulation | 6 |
| 2002 AMC 10A | Ratio of areas of two circles given equal arc lengths. | Geometry | Circular Geometry | 7 |
| 2002 AMC 10A | Compare areas of blue triangles, white squares, and the red square in a flag design. | Geometry | Triangles & Polygons | 8 |
| 2002 AMC 10A | Find the average of A, B, C from two linear equations with large coefficients. | Algebra | Systems & Algebraic Manipulation | 9 |
| 2002 AMC 10A | Sum of all roots of a factored quadratic equation. | Algebra | Polynomials | 10 |
| 2002 AMC 10A | Minimum number of 1.44 MB disks to store files of given sizes. | Algebra | Rates, Mixtures, & Averages | 11 |
| 2002 AMC 10A | Find speed needed to arrive exactly on time given late/early trips. | Algebra | Rates, Mixtures, & Averages | 12 |
| 2002 AMC 10A | Shortest altitude of a triangle with sides 15, 20, 25. | Geometry | Triangles & Polygons | 13 |
| 2002 AMC 10A | Number of possible k such that x^2 - 63x + k = 0 has prime roots. | Algebra | Polynomials | 14 |
| 2002 AMC 10A | Sum of four two-digit primes formed using given digits once each. | Number Theory | Factors, Multiples & Divisibility | 15 |
| 2002 AMC 10A | Find a+b+c+d from a chain of equal expressions. | Algebra | Systems & Algebraic Manipulation | 16 |
| 2002 AMC 10A | Fraction of cream in cup 1 after pouring coffee and cream back and forth. | Algebra | Rates, Mixtures, & Averages | 17 |
| 2002 AMC 10A | Smallest sum of visible pips on the surface of a 3x3x3 cube of dice. | Combinatorics | Counting | 18 |
| 2002 AMC 10A | Area accessible to a dog tethered outside a regular hexagon doghouse. | Geometry | Circular Geometry | 19 |
| 2002 AMC 10A | Ratio HC/JE given parallel segments and equidistant points on line AF. | Geometry | Triangles & Polygons | 20 |
| 2002 AMC 10A | Largest possible integer in an 8-integer set with mean, median, mode, range all 8. | Algebra | Rates, Mixtures, & Averages | 21 |
| 2002 AMC 10A | Number of operations to reduce 100 tiles to 1 by removing perfect squares. | Algebra | Sequences & Series | 22 |
| 2002 AMC 10A | Find AB such that perimeter of triangle AED is twice that of isosceles BEC. | Geometry | Triangles & Polygons | 23 |
| 2002 AMC 10A | Probability Sergio's number exceeds the sum of Tina's two distinct numbers. | Combinatorics | Probability | 24 |
| 2002 AMC 10A | Area of trapezoid with bases 39, 52 and legs 5, 12. | Geometry | Triangles & Polygons | 25 |
| 2002 AMC 10B | Simplify the quotient of exponentials: 2^{2001} * 3^{2003} / 6^{2002}. | Algebra | Logarithms & Exponents | 1 |
| 2002 AMC 10B | Evaluate the custom operation D(2,4,6) where D(a,b,c)=abc/(a+b+c). | Algebra | Functions | 2 |
| 2002 AMC 10B | Find which digit is missing from the 9-digit arithmetic mean of {9,99,...,999999999}. | Algebra | Sequences & Series | 3 |
| 2002 AMC 10B | Simplify (3x-2)(4x+1)-(3x-2)4x+1 and evaluate at x=4. | Algebra | Systems & Algebraic Manipulation | 4 |
| 2002 AMC 10B | Two externally tangent circles radii 2 and 3 inside a larger circumscribing circle; find the shaded area between them. | Geometry | Circular Geometry | 5 |
| 2002 AMC 10B | Find how many positive integers n make n^2-3n+2 a prime number. | Number Theory | Factors, Multiples & Divisibility | 6 |
| 2002 AMC 10B | Given 1/2+1/3+1/7+1/n is an integer, determine which statement about n is false. | Number Theory | Factors, Multiples & Divisibility | 7 |
| 2002 AMC 10B | If July has five Mondays, which day of the week must occur five times in August? | Number Theory | Modular Arithmetic | 8 |
| 2002 AMC 10B | Find the alphabetical position of USAMO among all 5-letter arrangements of A,M,O,S,U. | Combinatorics | Counting | 9 |
| 2002 AMC 10B | Find the pair (a,b) of nonzero real numbers that are roots of x^2+ax+b=0. | Algebra | Polynomials | 10 |
| 2002 AMC 10B | Three consecutive positive integers have product 8 times their sum; find the sum of their squares. | Algebra | Systems & Algebraic Manipulation | 11 |
| 2002 AMC 10B | Find k for which (x-1)/(x-2) = (x-k)/(x-6) has no solution for x. | Algebra | Systems & Algebraic Manipulation | 12 |
| 2002 AMC 10B | Find x such that 8xy-12y+2x-3=0 is true for all y. | Algebra | Systems & Algebraic Manipulation | 13 |
| 2002 AMC 10B | If N^2 = 25^64 * 64^25, find the digit sum of N. | Algebra | Logarithms & Exponents | 14 |
| 2002 AMC 10B | Positive integers A,B,A-B,A+B are prime; find the nature of their sum. | Number Theory | Factors, Multiples & Divisibility | 15 |
| 2002 AMC 10B | Find how many integers n make n/(20-n) a perfect square. | Number Theory | Factors, Multiples & Divisibility | 16 |
| 2002 AMC 10B | A regular octagon of side length 2; find area of triangle ADG. | Geometry | Triangles & Polygons | 17 |
| 2002 AMC 10B | Four distinct circles in a plane: maximum number of intersection points of at least two circles. | Combinatorics | Counting | 18 |
| 2002 AMC 10B | Arithmetic sequence with sums of first 100 and second 100 terms given; find common difference. | Algebra | Sequences & Series | 19 |
| 2002 AMC 10B | Given a-7b+8c=4 and 8a+4b-c=7, find a^2-b^2+c^2. | Algebra | Systems & Algebraic Manipulation | 20 |
| 2002 AMC 10B | Three mowers with different lawn sizes and speeds; who finishes first? | Algebra | Rates, Mixtures, & Averages | 21 |
| 2002 AMC 10B | Right triangle with midpoints; legs' segments to opposite vertices given; find hypotenuse. | Geometry | Triangles & Polygons | 22 |
| 2002 AMC 10B | Sequence defined by a_1=1 and a_{m+n}=a_m+a_n+mn; find a_12. | Algebra | Sequences & Series | 23 |
| 2002 AMC 10B | Ferris wheel radius 20 ft, 1 rev/min; time to go from bottom to 10 ft above bottom. | Geometry | Circular Geometry | 24 |
| 2002 AMC 10B | Adding 15 to a list increases mean by 2; adding 1 then decreases mean by 1; find original number of integers. | Algebra | Systems & Algebraic Manipulation | 25 |
| 2001 AMC 10 | Median of nine-number list is 10; find the mean. | Algebra | Rates, Mixtures, & Averages | 1 |
| 2001 AMC 10 | x is 2 more than product of its reciprocal and additive inverse; find interval containing x. | Algebra | Systems & Algebraic Manipulation | 2 |
| 2001 AMC 10 | Sum of two numbers is S; add 3 to each then double; find new sum. | Algebra | Systems & Algebraic Manipulation | 3 |
| 2001 AMC 10 | Maximum number of intersection points of a circle and a triangle. | Geometry | Circular Geometry | 4 |
| 2001 AMC 10 | How many of twelve pentominoes have at least one line of symmetry? | Geometry | Triangles & Polygons | 5 |
| 2001 AMC 10 | Two-digit N equals sum of product and sum of its digits; find units digit. | Number Theory | Factors, Multiples & Divisibility | 6 |
| 2001 AMC 10 | Decimal point moved four places right gives four times reciprocal; find original number. | Algebra | Systems & Algebraic Manipulation | 7 |
| 2001 AMC 10 | Four tutors work every 3,4,6,7 days; today all work; find next day all work together. | Number Theory | Factors, Multiples & Divisibility | 8 |
| 2001 AMC 10 | Tax rate p% on first $28000, (p+2)% on excess; total tax (p+0.25)% of income; find income. | Algebra | Rates, Mixtures, & Averages | 9 |
| 2001 AMC 10 | Given xy=24, xz=48, yz=72 with positive x,y,z; find x+y+z. | Algebra | Systems & Algebraic Manipulation | 10 |
| 2001 AMC 10 | Number of unit squares in the 100th ring around a center square. | Algebra | Sequences & Series | 11 |
| 2001 AMC 10 | Product of three consecutive integers divisible by 7; which choice is not necessarily a divisor? | Number Theory | Factors, Multiples & Divisibility | 12 |
| 2001 AMC 10 | Phone number digits decreasing in three parts with parity constraints; find A. | Combinatorics | Counting | 13 |
| 2001 AMC 10 | 140 tickets sold for $2001, some full price, some half price; find amount from full-price tickets. | Algebra | Rates, Mixtures, & Averages | 14 |
| 2001 AMC 10 | Street with parallel curbs 40 ft apart; crosswalk stripes 50 ft long, curb segment 15 ft; find stripe distance. | Geometry | Triangles & Polygons | 15 |
| 2001 AMC 10 | Mean of three numbers is 10 more than least, 15 less than greatest; median is 5; find sum. | Algebra | Rates, Mixtures, & Averages | 16 |
| 2001 AMC 10 | Cone formed from 252° sector of circle radius 10; find correct slant height and radius. | Geometry | 3D Geometry & Solids | 17 |
| 2001 AMC 10 | Plane tiled by squares and pentagons; find percent enclosed by pentagons. | Geometry | Triangles & Polygons | 18 |
| 2001 AMC 10 | Number of ways to choose four donuts from three types. | Combinatorics | Counting | 19 |
| 2001 AMC 10 | Regular octagon formed by cutting isosceles right triangles from a square side 2000; find side length. | Geometry | Triangles & Polygons | 20 |
| 2001 AMC 10 | Cylinder with diameter=height inscribed in cone of base diameter 10 and height 12; find cylinder radius. | Geometry | 3D Geometry & Solids | 21 |
| 2001 AMC 10 | Magic square with five entries unknown; find y+z. | Algebra | Systems & Algebraic Manipulation | 22 |
| 2001 AMC 10 | Draw chips without replacement until all red or all white drawn; probability last chip drawn is white. | Combinatorics | Probability | 23 |
| 2001 AMC 10 | Trapezoid with perpendicular bases, AB+CD=BC, AD=7; find product AB·CD. | Geometry | Triangles & Polygons | 24 |
| 2001 AMC 10 | Positive integers ≤2001 multiples of 3 or 4 but not 5; how many? | Number Theory | Factors, Multiples & Divisibility | 25 |
| 2000 AMC 10 | Positive distinct integers I, M, O with product 2001; find largest possible sum. | Number Theory | Factors, Multiples & Divisibility | 1 |
| 2000 AMC 10 | Simplify 2000(2000^2000) using exponent rules. | Algebra | Logarithms & Exponents | 2 |
| 2000 AMC 10 | Jenny ate 20% of jellybeans each day; after 2 days 32 remain. Find original number. | Algebra | Rates, Mixtures, & Averages | 3 |
| 2000 AMC 10 | Internet bill: fixed fee plus hourly charge; December $12.48, January $17.54 with twice the hours. Find fixed fee. | Algebra | Rates, Mixtures, & Averages | 4 |
| 2000 AMC 10 | Triangle PAB, M,N midpoints; P moves parallel to AB. How many of four quantities change? | Geometry | Triangles & Polygons | 5 |
| 2000 AMC 10 | Fibonacci sequence; which decimal digit appears last in the units position? | Number Theory | Modular Arithmetic | 6 |
| 2000 AMC 10 | Rectangle with AD=1, DB and DP trisect angle ADC; find perimeter of triangle BDP. | Geometry | Triangles & Polygons | 7 |
| 2000 AMC 10 | 2/5 of freshmen and 4/5 of sophomores took test; counts equal. Which ratio must hold? | Algebra | Rates, Mixtures, & Averages | 8 |
| 2000 AMC 10 | Given |x-2|=p with x<2, evaluate x-p. | Algebra | Systems & Algebraic Manipulation | 9 |
| 2000 AMC 10 | Triangles with sides 4,6,x and 4,6,y; find smallest positive number not a possible |x-y|. | Geometry | Triangles & Polygons | 10 |
| 2000 AMC 10 | Two different primes between 4 and 18; product minus sum equals which of given numbers? | Number Theory | Factors, Multiples & Divisibility | 11 |
| 2000 AMC 10 | Pattern of squares: figure 0 has 1, figure 1 has 5, etc. How many squares in figure 100? | Algebra | Sequences & Series | 12 |
| 2000 AMC 10 | Place 5 yellow, 4 red, 3 green, 2 blue, 1 orange peg on triangular board; no row/column same color. How many ways? | Combinatorics | Counting | 13 |
| 2000 AMC 10 | Five exam scores entered randomly; after each entry the average is integer. Find the last score entered. | Number Theory | Modular Arithmetic | 14 |
| 2000 AMC 10 | Non-zero reals a,b satisfy ab = a-b. Find possible value of a/b + b/a - ab. | Algebra | Systems & Algebraic Manipulation | 15 |
| 2000 AMC 10 | Lattice points; segment AB meets CD at E. Find AE length. | Geometry | Coordinate Geometry | 16 |
| 2000 AMC 10 | Coin machine: quarter→5 nickels, nickel→5 pennies, penny→5 quarters. Start with 1¢. Which amount possible? | Algebra | Systems & Algebraic Manipulation | 17 |
| 2000 AMC 10 | Walk around a 5km square; see 1km horizontally. Find area seen. | Geometry | Circular Geometry | 18 |
| 2000 AMC 10 | Right triangle, point on hypotenuse; lines parallel to legs form square and two small right triangles. Area ratio given, find other ratio. | Geometry | Triangles & Polygons | 19 |
| 2000 AMC 10 | Nonnegative integers A,M,C sum to 10. Maximize AMC+AM+MC+CA. | Algebra | Inequalities | 20 |
| 2000 AMC 10 | Alligators, ferocious creatures, creepy crawlers logical statements; which must be true? | Algebra | Logic & Truth Values | 21 |
| 2000 AMC 10 | Family drinks 8oz coffee+milk; Angela drinks 1/4 of total milk, 1/6 of total coffee. Find number of people. | Algebra | Rates, Mixtures, & Averages | 22 |
| 2000 AMC 10 | List: 10,2,5,2,4,2,x. Mean, median, mode form non-constant arithmetic progression. Find sum of possible x. | Algebra | Rates, Mixtures, & Averages | 23 |
| 2000 AMC 10 | Function with f(x/3)=x^2+x+1. Find sum of z such that f(3z)=7. | Algebra | Functions | 24 |
| 2000 AMC 10 | Year N: 300th day Tuesday; year N+1: 200th day Tuesday. Find day of week for 100th day of N-1. | Number Theory | Modular Arithmetic | 25 |
Andy and Betsy travel north and east from Mathville; find when they are equidistant from the start.
Two nut mixes combined to get 40% peanuts; find resulting weight of cashews.
Count isosceles triangles with integer sides and longest side 2025.
Students and teachers trivia contest; Ash changes average ages; find Ash's age.
Find the 2025th term of the sequence 1,2,1,2,3,2,1,...
Equilateral triangle trisectors form a convex hexagon; find its smallest angle.
Polynomial f(x)=x³+x²+ax+b; given remainders for x-1 and x-2, find b-a.
Agnes writes four self-referential statements; determine the number of false ones.
Given f(x)=100x³-300x²+200x, count a such that y=f(x-a) passes through (1,25).
Semicircle with chord CD of length 16 parallel to diameter AB; smaller semicircle removed; find shaded area.
Arithmetic sequence 1,x,y,z and geometric 1,p,q,z; z minimal integer; find x+y+z+p+q.
4-digit passcode with one even digit, one prime digit, no zero; count possible codes.
Infinitely nested squares alternately shaded; shaded area is 64% of original; find reduction ratio k.
Six chairs around a round table; 2 students and 2 teachers randomly sit; probability they sit adjacent in pairs.
Rectangle ABEF with lengths 7,1,5; AD perpendicular to DE; find area of triangle ABC.
Three coins randomly placed into three jars; expected number in the jar with the most coins.
N is the unique positive integer with 273436 mod N = 16 and 272760 mod N = 15; find tens digit of N.
Harmonic mean of all real roots of the product of 2025 quadratics of form kx²-4x-3.
Array built from -1,3,1 with adjacent sums; one row sums to 12288; find its third number.
Silo (diameter 20); MacDonald and McGregor positioned so line of sight is tangent; find distance g.
Find the maximum size of a sum-free subset of {1,2,...,20}.
Circle of radius r surrounded by three circles of radii 1,2,3 all externally tangent; find r.
Triangle ABC sides 80,45,75; angle bisector of B and altitude to AB intersect at P; find BP.
Count fair integers: distinct nonzero digits, no digit adjacent to two larger digits.
Point P inside square ABCD; probability AP is the middle length among AP, BP, AB.
Max mugs from 350g bag at 20g per mug
Sum of ones digits of squares 1^2 to 2025^2
Sum of digits of sum of 11th row in Pascal-like triangle
Two-digit base equality: ab (base7) = ba (base9), find sum a+b
Angle CAO in triangle with given sides and angle B=130
Area division of square by line, find equal-area line x=a
Distance paths to box via unlocked gates, equal distances
Cost $ABBA divisible by 36, find digit sum A+B
Count integer solutions to system of inequalities
Integers n such that rational function f(n)/g(n) is integer
Probability exactly 2 students same tutor both days
Compare area ratios of disks to enclosing polygons
Ratio of segments on altitude in 30-60-90 triangle
Probability tallest 3 athletes are group captains
Infinite sum of rational series in k
Distinct colorings of 6-sector circle with 2 of each color, no adjacent same
Max n for decreasing integer sequence with given average drop
Ones digit of sum of floor(√n) for n=1 to 2025
Fill time for remainder of container with trapezoidal sides
Diameter of circle tangent to four inscribed semicircles in square
Distinct colorings of 3x3 grid with cyclic adjacency constraint
Probability random 7-digit number with digit sum 61 is divisible by 11
Count squares with same number in two fill orders of 141x91 grid
Probability frog reaches 4 with given hop and disappear probabilities
Path from P to Q reflecting in square, find terminal vertex
Evaluate 9901·101 − 99·10101.
Hiking time model T=aL+bG; find T for given L and G.
Least prime that is sum of 5 distinct primes; sum its digits.
2024 as sum of two-digit numbers, least number of terms needed.
Least n such that n! is a multiple of 2024.
Minimum adjacent swaps to reverse string ABCDEF.
Three integers multiply to 60, least possible positive sum.
Chocolate factory packing rates; find when Daria joined.
6 juniors, 6 seniors into 3 teams of 2 each, count ways.
Apply operation (divide by 3 or add 10) 100 times starting at 100.
Integer solutions to sqrt(n²−49)=m; count ordered pairs.
Zelda's daily score changes over 6 days; find average score.
Which pairs of transformations (translate, rotate, reflect, dilate) commute?
Equilateral triangle, tangent circle; area of exterior region.
M+1213 and M+3773 are perfect squares; units digit of M.
Nested similar rectangles with given areas; find length AB.
Best-of-3 playoff probabilities; find away win probability p.
Bases b where 2024_b is divisible by 16; count and sum digits of count.
Geometric sequence a, 720, b; least b with sum of digits.
Subset of 1..2024 with spacing constraints; max possible size.
5×5 grid with arithmetic rows/columns; find missing entry.
Eight kites made of 30-60-90 triangles form polygon; area of triangle ABC.
Integers a,b,c satisfy three equations; find ab+bc+ca.
Bee moves on 3D grid via die rolls; probability of traversing four distinct edges of a unit cube.
Place toothpicks on 8×3 grid to form closed loop; count ways.
1013th from left is 1010th from right; find total number of people in line
Evaluate 10! - 7! * 6!
Count integer x with |2x| ≤ 7π
Balls placed in bins A-E cyclically with increasing count; find bin for ball 2024
Change + to – in sum of odds to make total negative; least number of changes
Rectangle integer sides area 2024; least possible perimeter
Remainder of 7^{2024}+7^{2025}+7^{2026} divided by 19
Product of all positive divisors of 42; find units digit
Arithmetic mean of a,b,c is 0, of squares is 10; find mean of ab,ac,bc
Parallelogram with midpoint E; intersection F of EB and AC; area ratio of CDEF to CFB
Rectangle with points M, A, right angle, equal triangle areas; find area of WMA
100 students, each speaks k languages, any pair lacks some common language; least total languages
√x+√y=√1183; positive integers x,y; minimize x+y
Probability dart in region (x²+y²-25)²≤49 inside |x|+|y|≤8
List of 9 numbers, given six, x≤y≤z, range 7, integer mean and median; count triples (x,y,z)
Repeatedly replace four numbers by sum/product; all remaining numbers odd; max possible count
5 snails race, at most one tie (any size); number of possible results
Possible remainders when an integer to the 100th power is divided by 125
For slopes 0, rational nonzero, irrational; which numbers of lattice points are possible
Three pairs of shoes; no left shoe next to right shoe from a different pair; number of arrangements
Two pipes radii 1 and 1/4 on floor; third pipe tangent to both and floor; sum of possible radii
16 people into 4 indistinguishable committees of 4, each with chair and secretary; count assignments, find exponent of 3
Sum of F₂/F₁ + F₄/F₂ + ... + F₂₀/F₁₀ for Fibonacci numbers
P(m)=m/2 + m²/4 + m⁴/8 + m⁸/8; how many of P(2022..2025) are integers
Bricks a×b×c form 3×3×3 block; reconfigured to 2×2×7 block 1 unit larger each dimension; find a+b+c
Alicia and Beth bike toward each other from cities 45 miles apart; find meeting distance from city A.
Weight of a large pizza given a relationship with orange slices and known slice weight.
Count positive perfect squares less than 2023 that are divisible by 5.
Greatest possible side length of a quadrilateral with integer sides, perimeter 26, and one side 4.
How many digits are in the base‑ten representation of 8⁵·5¹⁰·15⁵?
Value of a cube defined by vertex, edge, and face sums given the vertex sum is 21.
Probability that a running total of 4 die rolls equals 3 at some point.
Convert 200°F to the linear Breadus scale given two reference points.
For 2023, count dates YYYYMMDD where each digit appears an even number of times.
Maureen’s current quiz mean given how extra scores change the average.
Ratio of shorter to longer leg in a shaded right triangle between two squares of areas 2 and 3.
Number of three‑digit multiples of 7 whose digit reversal is a multiple of 5.
Square of the maximum distance from Abdul to Bharat given fixed angle and distance between Abdul and Chiang.
Probability that a randomly chosen divisor of a random integer 1–100 is divisible by 11.
Least even number of nested circles needed for total shaded area ≥ 2023π.
Total games played in a table tennis tournament given a player handedness ratio and a win ratio.
Perimeter of triangle APQ in a rectangle with integer side lengths for three right triangles.
How many vertices of a rhombic dodecahedron have exactly 3 edges meeting?
Center of rotation P(r,s) for a segment rotated to another; find |r−s|.
Number of colorings of a 3×3 grid with four colors such that every 2×2 block has all four colors.
Find m+n where the unique non‑integer root of a minimal polynomial is m/n.
Radius of the fourth circle internally tangent to two unit circles and externally tangent to a larger inscribed circle.
Sum of digits of N where N has two pairs of complementary divisors with differences 20 and 23.
Area inside a large regular hexagon not covered by six smaller regular hexagons, given a spacing of 3/7.
Probability that graph distance on a regular icosahedron satisfies d(Q,R) > d(R,S) for three random distinct vertices.
Equalizing orange juice amounts among four glasses by pouring.
Find original price of shoes given discount, tax, and total money.
Ratio of areas of circles circumscribing 3-4-5 and 5-12-13 right triangles.
Area covered by paint strip given width in mm, length in m, unit conversion.
Find number of terms in list given sums after adding 3 and multiplying by 3.
Number of even terms in recurrence L1=1, L2=3, L_{n+2}=L_{n+1}+L_n.
Angle between side of square and its copy rotated 20° about center.
Units digit of 2022^2023 + 2023^2022.
Number of consecutive perfect square pairs with difference at most 2023.
Minimum guesses to guarantee finding a hidden 2x1 rectangle on 3x3 grid.
Number of bill combinations ($20, $50, $100) totaling $800 with each denomination used.
Number of intervals where P(x) is positive after removing its 10 roots.
Area of region defined by ||x|-1|+||y|-1| ≤ 1.
Integer solutions (m,n) to m^2+mn+n^2 = m^2 n^2.
Smallest m making m·2!·3!·...·16! a perfect square.
Difference between number of upnos and downnos of at least two digits.
Longest interior diagonal of box given edge sum, face area sum, volume.
Which gcd statements are true given a/14 + b/15 = c/210?
Probability a frog jumping randomly from a point in a square lands outside.
Length of curve of four congruent semicircles drawn on a sphere of radius 2.
Probability 2023 balls placed randomly into 3 bins all have odd counts.
Number of solutions to floor(x)^2 - 3x + 2 = 0.
Find a+d+n for arithmetic sequence with sum off by 1 from 222.
Perimeter of parametric region (2u-3w, v+4w) for u,v,w in [0,1].
Area of smaller pentagon formed by folding vertices of a regular pentagon to center.
Evaluate the continued fraction 3 + 1/(3 + 1/(3 + 1/3)).
Mike cycles 15 laps in 57 min; estimate laps in first 27 min.
Find absolute difference of first and second numbers given sum and relationships.
Convert fuel efficiency from miles per gallon to liters per 100 km with general units.
Square side 1, equilateral convex hexagon inscribed, find side length s.
Simplify |a - 2 - √((a-1)^2)| for a < 0.
Find sum of digits of n given LCM(n,18)=180 and GCD(n,45)=15.
Mean of six numbers equals a value in set; find sum of possible X.
Color 5 regions of a partitioned rectangle with 5 colors, adjacent differ.
Rectangle with diagonal 8, cut 1x1 squares, distance 4√2, find area.
Equation 2^m √(1/4096) = 2 √[m](1/4096); find sum of real m.
Truth-tellers, liars, alternaters answer questions; candy given; find truth-tellers' candy.
Triangle with angle bisector AP, perpendicular BD, AD parallel BC; find AD.
Split integers 1-14 into 7 pairs where greater ≥ 2× lesser; count ways.
Area between circle and inscribed quadrilateral with sides 7,24,20,15.
Volume after lengthening each edge of box by 2; roots of 10x^3-39x^2+29x-6.
Three-digit with repeating decimal equation; count integer solutions.
Sequence of k° rotations and y-axis reflections returns (1,0); find least n.
Sum of reciprocals 1 to 17 equals h/L_17; find h mod 17.
Fourth term of sum of arithmetic and geometric sequences given first three sums.
Bowl of hexagons on square; find area of top octagon.
Cards 1-13 arranged; picked up in two passes; count orderings.
Isosceles trapezoid ABCD, point P with distances 1,2,3,4; find BC/AD.
Strings of digits 0-4 length 5 with at least j digits < j; count strings.
Squares R,S,T with lattice points conditions; find sum of edge lengths.
Evaluate expression with custom absolute difference operation.
Find area of rhombus given PB perpendicular to AD, AP=3, PD=2.
Count three-digit numbers with an odd number of even digits.
Find time of 700th hiccup starting at 4:00, occurring every 5 seconds.
Compute fraction involving products of fractions and a square root.
Determine how many of first ten terms of a specific sequence are prime.
Find number of k such that x^2+kx+36 has two distinct integer roots.
Find how many of 100 sets of 10 consecutive integers contain exactly two multiples of 7.
Sum the series of fractions n/(n+1)! from 1 to 2021.
Five positive integers with unique mode 2 more than median, median 2 more than mean; find least mode.
Logical equivalence of 'No school bigger than Euclid sold more T-shirts.'
Least n such that probability of rolling sum 7 at least once in n rolls > 1/2.
Two primes differ by 2, difference of cubes is 31106; find sum of digits of next prime.
Maximum size of subset of 1..25 where sum of any two elements is never in the set.
Given arithmetic sequence (d=2), S_{3n}/S_n constant; find S_20.
Rectangle 4×8, square side 5 placed on it; find area of overlap.
Which number among five given is not divisible by any prime less than 10.
How many systems of three linear equations with 0/1 coefficients have a nonzero solution.
Game of Life on 5×5 board; count initial configurations yielding center filled after one step.
Rhombus with angle 46°, midpoint, perpendicular; find angle BFC.
Find minimal-degree polynomial P(x) given remainders modulo x^2+x+1 and x^2+1; sum of squares of coefficients.
Sum of areas of all circles tangent to three given circles in the plane.
Ant Amelia: random increments and times uniform (0,1), up to 3 steps; probability position > 1.
Given Lipschitz condition and f(300)=f(900), maximize |f(f(800))-f(f(400))|.
Binary sequence x_k such that sum S_n satisfies 7S_n ≡ 1 mod 2^n; find weighted sum of four terms.
Evaluate the expression (2112-2021)^2 divided by 169.
Evaluate the expression (2^2-2)-(3^2-3)+(4^2-4).
Find new area after shortening the other side of a 4x6 card by 1 inch when shortening one side gave area 18.
Portia's school has three times as many students as Lara's; total 2600. Find Portia's students.
Maximum number of reshapable clay balls of radius 2 that fit in a cube of side 6.
Sum of two natural numbers is 17,402; one divisible by 10, erasing units digit gives other. Find difference.
How many minutes quicker is Route B than Route A given speeds and a school zone?
Cart accelerates down hill; distance each second increases by 7 inches. Find total distance in 30 seconds.
Find the digit A such that 20210A is prime.
Class mean of k>12 students is 8; 12 have mean 14. Find mean of remaining in terms of k.
Oscar's leap vs Elmer's stride length given strides between poles and total distance.
Chantal and Jean hike; given speeds, find Jean's average speed until they meet.
In a square with point E outside, find angle AFE given angle CDE = 110° and DE = DF.
Given statements about snakes (happy, can add, purple, can't subtract), which conclusion is valid?
Count two-digit cuddly numbers equal to tens digit plus units digit squared.
Multiplying 66 by a repeating decimal incorrectly gives answer 0.5 less. Find the two-digit repeating part.
Probability sum is even when rolling an unfair die twice (even 3x as likely as odd).
Find the least possible value of (xy-1)^2 + (x+y)^2 for real x and y.
t = average class size from teacher's view, s = from student's view; find t - s.
Which expression equals the product (2+3)(2^2+3^2)…(2^{64}+3^{64})?
Emily walks past a ship, counts steps; find length of ship in her steps.
For which base b is 2021_b − 221_b not divisible by 3?
Remainder when base-nine number 27,006,000,052_nine is divided by 5.
Two cones with equal liquid volume; marble dropped. Find ratio of liquid level rises.
Probability that each of 6 balls differs in color from more than half the others.
Find volume of tetrahedron with given edge lengths.
Number of ordered real pairs satisfying x^2+3y=9 and (|x|+|y|-4)^2=1.
Polynomial with positive integer roots has given form; find the coefficient B.
Area of circle through vertices A,B,C of isosceles triangle with tangent circle.
Select distinct A,B,C,D from 1-6; two parabolas intersect. How many selections?
Find the symmetry of f(x) = |floor(x)| - |floor(1-x)|.
List where integer n appears n times up to 200. Find the median of the list.
Height of pillar at E in a regular hexagon given heights at A, B, C support a flat solar panel.
Trapezoid with given side lengths and right angle; find AD length.
Number of ways to plant four crops in a 2x2 grid avoiding certain adjacent pairs.
Function f on rationals with f(ab)=f(a)+f(b) and f(p)=p for prime p. For which x is f(x)<0?
Side length s of square given swept areas of a disk rolling inside and outside.
Find area of region bounded by x^2+y^2 = 3|x-y|+3|x+y|.
Count positive integer pairs (b,c) so neither quadratic has two distinct real roots.
Number of rearrangements of 1..5 with no three consecutive increasing or decreasing terms.
Ratio p/q where p is probability of specific bin counts (3,5,4,4,4) and q of (4,4,4,4,4) when tossing 20 balls into 5 bins.
Equiangular hexagon; given areas of triangles formed by extending sides, find perimeter.
Radius r of three congruent spheres inside a right circular cone.
Hiram's 50-page notes on 25 sheets; after removing consecutive sheets, mean of remaining pages is 19. How many sheets removed?
Number of n ≤ 50 where iterating f_1 (twice divisor count) fifty times yields 12.
Frog on 3x3 grid with wrap-around; probability of reaching corner within 4 hops from center.
Label cube edges 0 or 1; count labelings where each face sum is 2.
Quadrilateral bounded by (x+ay)^2=4a^2 and (ax-y)^2=a^2; find its area in terms of a.
Value of the disrespectful quadratic at x=1 that maximizes sum of roots.
Place 3 red, 3 blue, 3 green chips on 3x3 grid with no two same color adjacent. How many ways?
Sum of four numbers formed by cyclically permuting digits 1,2,3,4
Count integer solutions to |x| < 3π
Area of shaded quadrilateral on a 6 by 5 grid
Simplify √((3-2√3)²)+√((3+2√3)²)
Compute numerator of fraction (2021/2020) - (2020/2021) in simplest form
Junior/senior debate team with percentages, find number of juniors
Temperature difference problem: Minneapolis N degrees warmer, find product of possible N after given changes
Pairs of blue/yellow shirts, find number of both-yellow pairs
Simplify n/4 where n=8^{2022} into a power of 4
Four distinct single-digit ages multiply to 24 and 30, find sum
Least positive integer with 2021 divisors expressed as m·6^k, find m+k
Mean scores of two classes with ratio 3:4, find overall mean
Count distinct integers that can be sum of two special fractions a/b with a+b=15
Maximize area of region inside exactly one of four tangent circles
Largest prime divisor of 16383; sum of its digits
Sum of greatest and least number in second row of spiral 15×15 grid
Fraction of red knights who are magical given ratios of colors and magical probability
Point rotated 90° around (1,5) then reflected over y=-x, find b-a
Alice and Bob draw numbers; dialogue leads to deduction that Bob has 2, Alice 25, sum 27
Water poured from cone into cylinder, find new height
Area of region bounded by six reflected arcs from sides of a regular hexagon inscribed in a circle
Brownie pan cuts, interior pieces equal perimeter pieces, maximize total
Find condition guaranteeing x(x-y)+y(y-z)+z(z-x)=1 for integers x,y,z
Ratio of sum of odd divisors to sum of even divisors of N
Area of isosceles triangle with two inscribed squares of side lengths 3 and 2
Base-n numeral 32d equals 263, find n+d given other numeral data
Probability that product of six dice is divisible by 4
Three parallel chords of circle lengths 38,38,34, find distance between adjacent lines
Square with points on sides, segments intersect at right angles, find area given segment lengths 6 and 7
Given x+1/x=√5, evaluate x^11−7x^7+x^3
Expected number of balls in original positions after two random adjacent swaps on a circle of five
Count uphill integers (strictly increasing digits) divisible by 15
Lines through origin, reflection of point yields another, find equation of second line
Five people receive two cards from 1-10; determine true statement about cards
Area of 24-sided polygon formed by three rotated squares of side 6
Probability every even number appears before first odd when rolling a fair die
Sum of leading digits of r-th root of a 313-digit repunit of 7s for r=2..6
Find average of all integers in S given average after removing max/min
Conditional probability that Hugo's first roll was 5 given he won the dice game
Area of pentagon built from 11 segments of length 2, express as √m+√n
Number of interior intersection points of sides of four regular polygons (5,6,7,8) inscribed in same circle
Folded square paper, find perimeter of triangle formed after fold
Sum of ten smallest n where sum of products jk for 1≤j<k≤n is divisible by 3
Probability at least one box gets three blocks same color among 5 colors
Probability that a monochromatic triangle exists when coloring sides and diagonals of a regular pentagon red/blue
Probability coin covers black region on square with corner triangles and diamond
Number of distinct 2x2x2 cubes with 4 white and 4 blue unit cubes up to rotation
Find brick-wall configuration where second player has a winning strategy
Sum of all possible areas of rectangle R inscribed in larger square with given smaller shapes
Lattice points 1≤x,y≤30 below line y=mx; length of m-interval for 300 points
Solve linear equation x - 3/4 = 5/12 - 1/3
Given average of five numbers, find average of two of them
Simplify product of three fractions involving a, b, c
Find net rate of pay per hour given distance, fuel efficiency, and pay rates
Sum of all real solutions to |x^2 - 12x + 34| = 2
Count 4-digit numbers with only even digits that are divisible by 5
Find common row sum of a 5x5 magic square using numbers -10 to 14
Evaluate alternating sum 1+2+3-4+...+197+198+199-200
Least number of bench sections for equal adults and children given capacities
Total surface area of a tower of seven stacked cubes with volumes 1^3 to 7^3
Median of concatenated list 1,...,2020 and 1^2,...,2020^2
Area of isosceles triangle with perpendicular medians of length 12
Probability a random-walking frog hits a vertical side of a 4x4 square
Evaluate algebraic expression given x+y=4 and xy=-2
Probability a randomly chosen divisor of 12! is a perfect square
Find distance d such that half of a square is within d of a lattice point
Number of integers n with P(n)<=0 where P(x)=∏(x-k^2)
Count ordered quadruples from {0,1,2,3} making ad-bc odd
Number of face paths from top to bottom of a regular dodecahedron
Area of quadrilateral with two right angles and given side lengths
Express (2^289+1)/(2^17+1) as sum of distinct powers of two
Count n≤1000 such that sum of floors of 998/n, 999/n, 1000/n not divisible by 3
Sequences of three isometries that return a given triangle to its position
Find smallest n>1000 with gcd conditions; sum its digits
Probability optimal reroll strategy uses exactly two dice in a die game
Evaluate 1-(-2)-3-(-4)-5-(-6)
Total volume of 5 cubes side 1 and 5 cubes side 2
Given w:x=4:3, y:z=3:2, z:x=1:6, find w:y
Right triangle with prime acute angles, find least possible smaller prime
Arrange 1 brown, 1 purple, 2 green, 3 yellow tiles in a row
Odometer palindrome 15951, find average speed to next palindrome in 2 hours
Count even multiples of 3 less than 2020 that are perfect squares
Locations for R such that triangle PQR right with area 12, PQ=8
Integer pairs (x,y) satisfying x^2020 + y^2 = 2y
Three-quarter sector of circle radius 4 rolled into cone, find volume
Probability two people pick exactly 2 same books out of 10, each picking 5
Number of leading zeros in decimal expansion of 1/20^20
Andy ant moves 1,2,3,... turning left, location at 2020th turn
Shaded area inside hexagon outside six semicircles on sides
Erasing every 3rd,4th,5th digit from repeated 12345, sum at positions 2019-2021
Bela and Jenn choose numbers >1 apart on [0,n], who wins?
Pair 10 people in circle into 5 mutually-knowing pairs
Urn with 1 red,1 blue; add same color ball 4 times, probability 3 of each
Find missing digit A in number 158A00A4AA0 equal to C(52,10)
Volume of points within distance r of a 1x3x4 prism, find bc/ad
Square with points E,F,G,H,I,J, all regions area 1, find FI^2
Remainder of 2^202+202 divided by 2^101+2^51+1
Count sequences of 20 transformations L,R,H,V returning square to original
Number of positive integers n with (n+1000)/70 = floor(sqrt(n))
Number of ordered factorizations of 96 into factors >1
Evaluate 2^(0^(1^9)) + ((2^0)^1)^9.
Find the hundreds digit of 20! − 15!.
Given ages last year and this year, find the age difference of Ana and Bonita.
Minimum balls drawn to guarantee 15 of one color from a box of mixed colors.
Greatest number of consecutive integers whose sum equals 45.
How many listed quadrilateral types have a point equidistant from all four vertices.
Area of triangle enclosed by two lines through (2,2) and the line x+y=10.
Number of rigid motions that map an infinite square-and-segment pattern onto itself.
Greatest three-digit n such that sum of first n integers does not divide n!.
Number of 1-foot tiles visited by a diagonal bug on a 10 by 17 foot floor.
Number of divisors of 201^9 that are perfect squares or perfect cubes.
Compare the mean, median, and median of the modes of 2019 calendar dates.
In isosceles triangle ABC with circle on BC, find angle BFC formed by diagonals.
Sum of all possible numbers of distinct intersection points of four lines.
Find p+q for a recursively defined sequence term a_{2019}=p/q.
Area inside a large circle that surrounds 13 tangent unit circles.
Number of distinct towers of height 8 using 2 red, 3 blue, 4 green cubes.
Find base k such that 0.232323..._k equals 7/51.
Minimum value of (x+1)(x+2)(x+3)(x+4)+2019 for real x.
Probability that each row and column sum is odd when numbers 1-9 fill a 3x3 grid.
Distance from a sphere's center to a triangle plane when triangle sides are tangent to sphere.
Probability |x-y| > 1/2 when numbers are chosen by coin flips or uniformly.
2019th number spoken by Tadd in a rotating counting game with increasing counts.
Evaluate 1/A + 1/B + 1/C from partial fraction decomposition of cubic reciprocal.
How many integers n in 1..50 make (n^2−1)!/(n!)^n an integer.
Alicia pours water between two containers; find ratio of volumes.
Find counterexample to 'if n not prime then n-2 is prime'.
Find number of non-seniors playing instrument given percentages.
Find common point of lines with coefficients in arithmetic progression.
Determine which statement about reflected triangle is not always true.
Solve factorial equation for n and find digit sum.
Find minimum n such that n purple candies cost equals cost of 12,14,15 other candies.
Find shaded area inside square with four equilateral triangles.
Find range of function with floor and absolute value.
How many points C make triangle with given perimeter and area?
Find difference in blue marbles between two jars given ratios and totals.
Find maximum digit sum in base-7 representation of number less than 2019.
Find x such that median equals mean of set {4,6,8,17,x}.
Find sum of missing digits in 19! using divisibility rules.
Two right triangles sharing two sides; find square of product of third sides.
Find ratio AD:DB in right triangle with given equalities.
Probability red ball in higher bin than green ball given geometric distribution.
Henry walks 3/4 distance repeatedly; find difference between limiting points.
Count distinct products of two distinct divisors of 100,000.
Find shaded area inside circle overlapping three semicircles.
Probability of getting two heads in a row with second tail seen before second head.
Probability three players each have $1 after 2019 rounds of random giving.
Find area of circle given two points and tangents intersecting on x-axis.
Find interval containing least m with x_m ≤ 4+1/2^20 for recursive sequence.
Count binary sequences length 19 with given restrictions.
Evaluate a nested expression with negative exponents and fractions.
Compare soda amounts given percent relationships among three people.
Find the date when 10! seconds after noon Jan 1 elapses.
Schedule 3 distinct math courses in 6 periods with no two consecutive.
Determine possible distance d given three false statements about the distance.
Find total votes given 65% likes and a score of 90.
Count integer n making 4000*(2/5)^n an integer.
Solve coin word problem with 23 coins, total 320 cents, find difference.
Find area of trapezoid in diagram of similar isosceles triangles.
Given sqrt(49-x^2)-sqrt(25-x^2)=3, find sqrt(49-x^2)+sqrt(25-x^2).
Find numerator n of probability that sum of 7 dice is 10.
Count solutions to system with absolute values: x+3y=3, ||x|-|y||=1.
Find crease length when folding a 3-4-5 triangle so A meets B.
Find floor of (3^100+2^100)/(3^96+2^96).
Find distance AB between tangency points of two circles inside a larger circle.
Count integer-length segments from right-angle vertex to hypotenuse.
Find least possible element of a 6-element subset of {1..12} with no multiples.
Count nonnegative integers expressible as sum a_i*3^i with a_i in {-1,0,1}.
Find probability m^n ends in 1; m from {11,13,15,17,19}, n from 1999-2018.
Count symmetric 7x7 scanning codes with at least one square of each color.
Find a for circle x^2+y^2=a^2 and parabola y=x^2-a intersecting at exactly 3 points.
Given GCD constraints among a,b,c,d, identify divisor of a.
Find fraction of right triangular field planted, leaving a square corner unplanted.
Find area of quadrilateral FDBG in triangle with midpoints and angle bisector.
Maximize a+b+c for C_n-B_n=A_n^2 with repeated-digit numbers, at least two n.
How many 2-inch squares from a 20x18 inch cornbread?
Average speed in last 30 min of a 90-min trip given first two segments.
Different values of (a×b)+(c×d) using digits 1-4 once.
Find X+Y+Z for box with face areas 24,24,48,48,72,72.
Subsets of {2,...,9} containing at least one prime.
Probability 3 draws needed until sum of chips exceeds 4.
Number of small semicircles on diameter given area ratio 1:18.
Steps in toothpick staircase using 180 toothpicks.
Sum with same probability as 10 when rolling seven dice.
Volume of pyramid with apex at midpoint of edge in rectangular box.
Which expression p^2+... is never prime for prime p?
Area traced by centroid of triangle on circle diameter AB=24.
Count of numbers 10^n+1 divisible by 101 in first 2018 terms.
Minimum distinct values in 2018 numbers with unique mode appearing 10 times.
Area of square wrapping paper folded over a box with base w and height h.
Sum of cubes mod 6 of strictly increasing sequence summing to 2018^{2018}.
Side length of equilateral octagon inscribed in rectangle of width 8, height 6.
Seating arrangements of 3 sibling pairs in a van with restrictions.
Sum of digits of Joey's age when his age next multiple of Zoe's.
Find f(2018) for recursive sequence f(n)=f(n-1)-f(n-2)+n.
Smallest next divisor after 323 of an even 4-digit number.
Probability that x, y, 1 form an obtuse triangle for x,y in [0,1].
Ordered pairs (a,b) satisfying equation with lcm and gcd.
Area of intersection of two triangles in a regular hexagon of side 1.
Number of real solutions to x^2+10000⌊x⌋=10000x.
Value of (2(2(2(2(2(2+1)+1)+1)+1)+1)+1)
Maximize popsicles bought with $8 given single $1, 3-pack $2, 5-pack $3
Total walkway area around six 6x2 ft flower beds in a 3x2 grid with 1-ft walkways
Mom puts 3 toys in box per 30 sec, child removes 2; time to put all 30 toys in box
Sum of two numbers equals 4 times their product; find sum of reciprocals
If all multiple choice right then get A; which statement logically follows?
Jerry walks two sides of square, Silvia diagonal; percent shorter is Silvia's trip?
30 people: 20 know each other, 10 know no one; hugs for known, handshakes for unknown; count handshakes
Minnie and Penny travel same route opposite ways with different speeds; time difference
Choose fourth rod from 1-30 to form quadrilateral with 3,7,15; how many choices?
Find length of segment AB given volume of region within 3 units of AB is 216π
Set S of (x,y) such that two of {3, x+2, y-4} equal and third ≤; describe S
Sequence F_n mod 3; sum of 8 consecutive terms starting F_2017
Ticket costs 20% of (allowance - soda), soda 5% of (allowance - ticket); fraction spent
Chloe picks x in [0,2017], Laurent y in [0,4034]; probability y > x
Find smallest T where at least 5 of 10 horses (lap times 1..10) meet at start
Points P,Q,R,S on circle x^2+y^2=25 with integer coords; maximize irrational distance ratio PQ/RS
Amelia and Blaine alternate coin tosses, different head probabilities; Amelia wins probability
Seat 5 people in a row: Alice not next to Bob/Carla, Derek not next to Eric
S(n) is digit sum; S(n)=1274; which could be S(n+1)?
Two squares inscribed in 3-4-5 right triangles: one at right angle, one on hypotenuse; ratio of sides
In equilateral triangle, sides AB, AC tangent to circle at B,C; fraction of area outside circle
How many triangles with vertices at integer coordinates (i,j) for 1≤i,j≤5?
Polynomial g has 3 distinct roots, each root of f; find f(1)
How many 3-digit numbers have a permutation that is a multiple of 11?
Mary multiplies two-digit number by 3, adds 11, reverses digits; find original number.
Sofia runs 5 laps with varying speeds; find total time.
Given inequalities for real numbers x, y, z; determine which expression is always positive.
Given fractional equation, find value of another fractional expression.
Jellybean count problem: find original number of blueberry given ratios after eating.
Maximum number of solid 2x2x1 blocks that fit in a 3x2x3 box.
Samia bikes and walks equal distances; find distance walked given total time.
Find coordinates of point C given triangle vertices A, B, altitude foot D, and AB=AC.
Contestant guesses on 3 multiple-choice questions; probability of at least 2 correct.
Two perpendicular lines intersect at (1,-5); find constant c.
Survey: find fraction of students who say they dislike dancing but actually like it.
Car fuel efficiency and price comparison; find percent savings.
Students taking classes; given numbers in each and overlap, find number taking all three.
Random N from 1 to 2020; find probability N^16 mod 5 is 1.
Rectangle ABCD, E foot from B to AC; find area of triangle AED.
Count positive integers ≤2017 containing digit 0.
Count monotonous numbers (digits strictly increasing or decreasing).
Paint 6 disks 3 blue, 2 red, 1 green; rotations/reflections considered same; count paintings.
Equilateral triangle extended; find ratio of areas of new triangle to original.
Random divisor of 21! chosen; probability it is odd.
Right triangle with D midpoint of hypotenuse; find sum of inradii of two smaller triangles.
Circle diameter extended, perpendicular drawn; find area of triangle formed.
N is integer 123...44; find remainder upon division by 45.
Equilateral triangle with vertices on xy=1 and centroid at hyperbola vertex; find squared area.
Isabella's 7 test scores distinct 91-100, averages integers; 7th=95; find 6th.
Evaluate the expression (11! - 10!) / 9!
Find x such that 10^x * 100^(2x) = 1000^5
Ben spent $12.50 more than David on bagels, with David spending 25¢ less per dollar; find total spent
Evaluate the remainder function rem(3/8, -2/5) using the given definition with floor
Rectangular box with side ratio 1:3:4 has integer volume; which given volume is possible?
Ximena lists 1..30, Emilio replaces each '2' digit with '1'; find difference in sums
Given 7 data values where mean, median, and mode all equal x, find x
Trickster Rabbit doubles Fox's money each crossing minus 40 coins; after 3 crossings money is gone; find starting amount
Triangular array of 2016 coins, N rows; find sum of digits of N
Rug with three concentric rectangles, areas in arithmetic progression; find inner rectangle length
Find the area of the shaded L-shaped region in a 8×5 rectangle
Probability that product of three distinct integers from 1 to 2016 is odd
Five friends in a row of seats move; find Ada's original seat given constraints
Number of ways to write 2016 as a sum of twos and threes, ignoring order
Seven radius-1 cookies cut from a larger circle; find the radius of the scrap cookie
Triangle reflected across x-axis then rotated 90° CCW; which transformation returns it to original?
Probability that at least 3/5 of N green balls are on same side of a red ball; find smallest N with P<321/400
Assign numbers 1..8 to cube vertices so each face sums the same; count distinct arrangements
Rectangle ABCD with points E,F on BC; find ratio BP:PQ:QD where AE,AF intersect BD
Expansion of (a+b+c+d+1)^N has exactly 1001 terms with all four variables; find N
Three circles tangent to a line and each other; find area of triangle formed by their centers
110n³ has exactly 110 divisors; how many divisors does 81n⁴ have?
Binary operation ◊ with given properties; solve 2016 ◊ (6 ◊ x) = 100 for x
Quadrilateral inscribed in circle of radius 200√2 with three sides 200; find the fourth side
Ordered triples of positive integers with given pairwise LCMs: lcm(x,y)=72, lcm(x,z)=600, lcm(y,z)=900
Evaluate (2a^{-1}+a^{-1}/2)/a for a=1/2
Evaluate (2♡4)/(4♡2) for custom operation n♡m=n^3 m^2
Evaluate nested absolute value expression with x=-2016
Zoey reads 15 books, each takes 1 more day, finished 1st on Mon, 2nd Wed, find day of 15th
Given mean and median of 4 cousins' ages, find sum of youngest and oldest
Two three-digit numbers with all six digits distinct, sum is three-digit, minimize digit sum of sum
Two acute angles ratio 5:4, complement of one twice complement of other, find sum
Find tens digit of 2015^{2016}-2017
Triangle vertices on y=x^2, A at origin, base parallel to x-axis, area 64, find base length
Weight of equilateral triangle side 3 is 12 oz, find weight of similar triangle side 5 (same thickness)
20 fence posts, corners, rest evenly spaced 4 yards, longer side has twice as many posts as shorter, find area
Two different numbers from {1,2,3,4,5} multiplied, probability product even
1000 babies from twins, triplets, quadruplets; ratios given, find how many were in quadruplet sets
Number of axis-aligned integer-coordinate squares inside region bounded by y=πx, y=-0.1, x=5.1
Numbers 1-9 in 3x3 grid, consecutive numbers share an edge, corners sum 18, find center number
Infinite geometric series with second term 1, find minimum possible sum S
Numbers 2-7 on cube faces, each vertex product of three adjoining faces, maximize sum of eight products
Number of ways to write 345 as sum of two or more consecutive positive integers
Rectangle ABCD with points E,F,G, segments AG,AC intersect EF at Q,P; find PQ/EF
Dilation sends circle radius 2 center (2,2) to radius 3 center (5,6); find distance origin moves
Area enclosed by graph of x^2+y^2=|x|+|y|
21 teams, each won 10 lost 10; number of 3-team cycles A beats B beats C beats A
In regular hexagon ABCDEF, points W,X,Y,Z on sides; lines AB,ZW,YX,ED parallel and equally spaced; find area ratio of WCXYFZ to entire hexagon
Four-digit integers abcd where two-digit numbers ab, bc, cd form increasing arithmetic sequence; count how many
f(x)=sum_{k=2}^{10} (floor(kx)-k floor(x)), count distinct values for x≥0
Simplify (2^0-1+5^2-0)^{-1} times 5
Box of 25 triangular and square tiles with 84 edges, find number of square tiles
Add toothpicks to extend a 3-step staircase to a 5-step staircase
Equalize eggs; Pablo 3x Sofia, Sofia 2x Mia; fraction of Pablo's eggs to Sofia
Class average 80 without Payton, 81 with; find Payton's score
Sum of two numbers is 5 times their difference; find ratio of larger to smaller
Number of terms in arithmetic sequence 13, 16, 19, ..., 73
Find years until Pete is twice Claire, given ages two and four years ago
Two cylinders have equal volume, second radius 10% larger; compare heights
Rearrangements of abcd with no two adjacent letters also adjacent in alphabet
Rectangle length:width=4:3, diagonal d, area kd^2; find k
Given points (√π, a) and (√π, b) on x^4+y^2=2x^2y+1, find |a-b|
12 coins each 5¢ or 10¢, exactly 17 different totals; number of 10¢ coins
Disk rolls clockwise around clock; when does arrow next point upward?
Relatively prime fraction increases by 10% when both terms increased by 1
Solve symmetric system y+4=(x-2)^2, x+4=(y-2)^2 with x≠y; find x^2+y^2
Equilateral triangle from x=1, y=1+√3/3 x, and a line through origin; perimeter
Among first 1000 integers, count those with only numeric digits in hexadecimal
Isosceles right triangle area 12.5, trisect right angle; area of middle triangle
Rectangle integer sides; which of 100,102,104,106,108 cannot equal area+perimeter?
Tetrahedron with AB=5, AC=3, BC=4, BD=4, AD=3, CD=12√2/5; find volume
8 people flip fair coins; probability no two adjacent people stand
Zeros of x^2-ax+2a are integers; sum of possible values of a
Quadrilateral with AB=2, right angles B,C, CD=AD, integer sides, count p<2015
Two random points on sides of unit square; probability distance is at least 1/2
Evaluate 2 minus (-2) to the -2 power
Marie finishes three equal tasks, find finish time of third task
Sum of two numbers twice and three times is 100, one is 28, find the other
Four siblings eat fractions of a pizza, order by amount consumed
Six-person race with position constraints, find who finished 8th
Mahdi's sport schedule with constraints, find the day he swims
Evaluate nested custom operation a diamond b = a - 1/b
Letter F rotated 90° clockwise, reflected in y-axis, half turn; find final image
Area of shark's fin shape bounded by quarter-circles and line segment
Product of odd negative integers > -2015, find sign and units digit
Probability a number with prime digits is prime, selected from numbers less than 100
Integer points (x,-x) inside or on circle radius 10 centered at (5,5)
Sum of altitudes of triangle formed by line 12x+5y=60 and axes
Maximize sum of roots of (x-a)(x-b)+(x-b)(x-c)=0 for distinct one-digit a,b,c
Given ratios of people, horses, sheep, cows, ducks, find impossible total count
Probability Al, Bill, Cal numbers are multiples with distinct assignments 1-10
Volume of octahedron formed by joining face centers of 3x4x5 prism
Expected number of heads after flipping coins and retossing tails up to three times
Right triangle ABC with squares, X,Y,Z,W concyclic, find perimeter
Ant on cube visits all corners exactly once in 7 moves, can't return; count paths
Staircase with jumps of 2 and 5 steps, Dash takes 19 fewer jumps than Cozy
Regular pentagon with diagonals, AG=1, find FG+JH+CD
n! ends in k zeros, (2n)! ends in 3k zeros, find sum of four smallest n
Ant walks spiral path on coordinate plane, find p_2015
Rectangular box with integer dimensions, volume equals surface area, count ordered triples
Evaluate 10*(1/2+1/5+1/10)^{-1} and simplify.
Cat eats 1/3 can in morning and 1/4 in evening; after how many days is 6-can box finished starting Monday morning?
Bridget sells 48 loaves at different prices during the day; find total profit given cost per loaf.
Four painted houses in a row with color order constraints; count possible arrangements.
Percentages of quiz scores given; find difference between mean and median.
a cows give b gallons in c days; find gallons from d cows in e days.
Given x < a and y < b, determine how many inequalities must be true.
Which of five factorial expressions is a perfect square?
Right triangle legs 2√3 and 6; find length of third altitude.
Five positive consecutive integers starting with a have average b; find average of five integers starting with b.
Customer has three coupons; for which listed price does coupon 1 give the greatest discount?
Regular hexagon side 6 with six 120° sectors of radius 3; find shaded area inside hexagon.
Equilateral triangle side 1 with outward squares on each side; find area of hexagon formed.
Two perpendicular lines intersect at (6,8) and have y-intercepts summing to zero; find area of triangle formed.
David drives 35 miles first hour, then speeds up; arrives 30 min early instead of 1 h late; find total distance.
Rectangle ABCD with midpoints; shade region and find its area.
Three fair dice rolled; probability that two numbers sum to the third number.
Square has y-coordinates 0,1,4,5; find its area.
Stacked cubes of edge 1,2,3,4; find length of segment XY inside cube 3.
Product 8 * (888...8) with k digits has digit sum 1000; find k.
Lines y=ax+5 and y=3x+b intersect x-axis at same point; a,b positive integers; find sum of all possible x-coordinates.
Rectangle AB=20, BC=10, E on CD with angle CBE=15°; find AE.
Rectangular paper length=√3 times width, folded along dotted line; find ratio B/A of folded area to original.
Sequence built by listing n+3 numbers then skipping n; find 500,000th term.
Given 5^867 between 2^2013 and 2^2014; count pairs (m,n) with 1≤m≤2012 satisfying 5^n<2^m<2^{m+2}<5^{n+1}.
Leah has 13 coins; with one more nickel, pennies equal nickels. Find total value.
Simplify (2^3+2^3)/(2^{-3}+2^{-3}) to a number.
Randy drives first third on gravel, 20 miles on pavement, one-fifth on dirt. Find total trip length.
Susie buys 4 muffins and 3 bananas; Calvin buys 2 muffins and 16 bananas for twice the cost. Find muffin-to-banana cost ratio.
A square window made of 8 equal panes (5:2 ratio) and 2-inch borders. Find the side length of the window.
Orvin can buy 30 balloons normally; with 'buy one, second 1/3 off', how many can he buy?
If A > B > 0 and A is x% greater than B, express x in terms of A and B.
A truck travels b/6 feet every t seconds. How many yards in 3 minutes?
Given a complex fraction equals 2014, find (w+z)/(w-z).
Cryptarithm: ABBCB+BCADA=DBDDD with distinct digits; how many possible D values?
Find smallest integer n% single discount better than two 15% off, three 10% off, or 25% then 5% off.
Find the fifth-largest divisor of 2,014,000,000.
Six regular hexagons surround a hexagon of side 1; find area of triangle ABC formed.
Danica drives whole hours at 55 mph; odometer start abc end cba with digit sum ≤7. Find a^2+b^2+c^2.
Rectangle with DC=2BC, E,F trisect angle ADC; find ratio of area triangle DEF to rectangle ABCD.
Four fair dice rolled; probability at least three show same value.
Find the greatest power of 2 dividing 10^{1002} - 4^{501}.
11 positive integers, mean 10, median 9, unique mode 8; maximize the largest integer.
Two concentric circles radii 1 and 2; two random points on outer circle; probability chord intersects inner circle.
How many integers x make x^4-51x^2+50 negative?
Trapezoid ABCD, AB||CD, AB=33, CD=21, legs 10 and 14; acute A and B; find shorter diagonal.
Eight semicircles inside a square of side 2; find radius of circle tangent to all.
Sphere inscribed in truncated cone; volume of frustum twice sphere; find ratio of bottom to top radius.
Numbers 1-5 in a circle; arrangement is bad if not all sums 1-15 achievable by consecutive numbers. Count bad arrangements (mod rotation/reflection).
Frog on lily pads 1, escapes to 10, eaten at 0; transition probabilities based on pad. Start pad 1. Find escape probability.
Find cost of a 5-mile taxi ride given base fare and per-mile rate.
How many 1/4-cup measures to make 2 1/2 cups of sugar?
Square of side 10, point E on BC, area of triangle ABE = 40; find BE.
Softball team scores 1..10 runs; deduce opponent runs and find total.
Split vacation costs evenly; find difference t-d after reimbursements.
Deduce Kris's age from sibling activities and age constraints.
Choose 4 courses with English and at least one math; count ways.
Simplify fraction with large powers of 2.
Basketball: 20% of 3-pt, 30% of 2-pt successful on 30 attempts; points?
Bouquet of roses and carnations; given fractions, find percent carnations.
Council: 10 ways to choose 2-person committee; how many for 3-person?
Isosceles triangle with parallel lines; find perimeter of parallelogram.
Count three-digit numbers not divisible by 5, digit sum <20, first=last digit.
Solid cube 3x3x3 with unit cubes removed from each corner; count edges.
Triangle sides 10, 15; altitude to third side is average of other altitudes.
Triangle reflected across x=8; find area of union of original and reflected.
Alice, Beatrix, Claire visit every 3,4,5 days; count days exactly two visit.
Quadrilateral cut into equal-area halves by line through A; find intersection.
Count bases b>3 such that 2013 ends in digit 3 in base b.
Area swept out by rotating a unit square 45° about its center.
Pirates take fraction of remaining gold; smallest initial to give integer shares.
Six unit spheres at hexagon vertices inside larger sphere; find 8th sphere radius.
Triangle ABC, circle center A radius AB meets BC at X; integer BX, CX; find BC.
Schedule 6 rounds of 3-on-3 backgammon with each pair playing twice; count ways.
All diagonals drawn in a regular octagon; count interior intersection points.
Evaluate the expression (2+4+6)/(1+3+5) - (1+3+5)/(2+4+6)
Find pounds of potatoes from garden based on step measurements
Find low temperature given high is 16 more and average is 3
Determine the nth number when counting backwards from 201 to 3
Minimize 2a - ab given a,b positive integers less than 6
Find overall average age of fifth-graders and parents
Find area of triangle formed by three equally spaced points on a circle
Find combined miles per gallon for two cars driving same distance
Sum of three pairwise relatively prime integers with product 27000
Find number of three-point shots attempted given percentages and totals
Find x+y satisfying x^2+y^2=10x-6y-34
Probability two chosen segments from a regular pentagon have same length
Find the 53rd number spoken in a turn-taking counting game
Describe set of points where a custom operation is commutative
Find ratio a/b when equilateral triangle and regular hexagon have equal area
Find area of quadrilateral AECD given medians and segment lengths in triangle ABC
Find maximum silver tokens from exchange booths with red and blue tokens
Count integers 1000-2013 where units digit equals sum of other digits
Find the double root of quadratic with coefficients in arithmetic progression
Find |a1-b1| when 2013 expressed as ratio of factorials with minimal a1+b1
Smallest N from two sequences with same seventh term
Number of ways to assign digits to octagon vertices and center with equal line sums
Find length DF in triangle with perpendiculars and given side lengths
How many numbers 2010-2019 are 'nice' (sum of four divisors of m)
Count N with last two digits of base-5 and base-6 sum matching 2N
Cagney frosts a cupcake every 20 s, Lacey every 30 s; together how many in 5 min?
A square of side 8 is cut in half; find dimensions of a rectangle.
Bug crawls from -2 to -6 to 5 on number line; total units crawled.
Angles 24° and 20° sharing ray; smallest possible third angle.
100 adult cats, half female, half have litters of 4 kittens; total cats and kittens.
Two positive numbers product 9, one reciprocal is 4 times other; find sum.
Bag 3/5 blue, rest red, double red; fraction red after doubling.
Three numbers pairwise sums 12,17,19; find the middle number.
Dice: one even faces only, one odd faces only; probability sum is 7.
Circle split into 12 sectors with integer angles in arithmetic sequence; smallest angle.
Two externally tangent circles radii 5 and 3, common external tangent; find BC.
Leap year rules; 200th birthday Tuesday, find day of week born.
Iterative average of 1,2,3,4,5; difference between max and min possible values.
31x31 checkerboard black corners; how many black squares?
Three unit squares and two segments connecting vertices; area of triangle ABC.
Three runners on 500 m track speeds 4.4,4.8,5.0; time until all together.
Relatively prime a,b with fraction condition; find a-b.
Closed curve of 9 arcs from regular hexagon centers; enclosed area.
Paula and two helpers paint house with lunch break; break length in minutes.
3x3 grid random colors, rotate 90° and recolor; probability all black.
Points (0,0,0),(1,0,0),(0,2,0),(0,0,3); midpoints form quadrilateral area.
Sum of first m odd integers is 212 more than sum of first n even; sum of all possible n.
Six students, each same number of friends; total number of friendship arrangements.
Integers a≥b≥c>0 satisfying two given equations; find a.
Real numbers x,y,z in [0,n]; probability none within 1 > 1/2, smallest integer n.
Students and rabbits per classroom, total difference across four rooms
Circle inscribed in a 2:1 rectangle, find area of rectangle
Reflect point (1000,2012) across y=2000, find new coordinates
Leftover marbles after packing 6 per bag, find combined leftovers
Dinner cost with 10% tax and 15% tip on pre-tax, find original price
Effect of rounding x up and y down on estimate of x-y
Animals hiding acorns, different per hole, same total, find number
Sum of integer solutions to compound quadratic inequality
Minimum odd integers among six with given consecutive sums
Count ordered pairs of positive integers solving M/6 = 6/N
Dessert menus for a week with no repeat adjacents and cake on Friday
Points A, B, C, D with directions and distances, estimate AD
Escalator: walking vs standing times, find ride-down time
Two equilateral triangles in a square overlap as a rhombus, find area
Round-robin 6 teams, maximum teams tied for most wins
Three mutually tangent circles radius 2, total area of circles and inner region
Paper disk cut into two sectors forming cones, volume ratio smaller to larger
False positive test probability, find P(disease | positive test)
Rectangle with extended side, intersection and quadrilateral area
Game of doubling and adding 50, smallest starting number for Bernardo to win
Four points, segment lengths a,a,a,a,2a,b, find b:a ratio
List of first 10 integers with adjacent-preview condition, count lists
Tetrahedron cut from unit cube, remaining shape height on face
Three girls like four songs with pairwise-disjoint-like constraints, count ways
Bug traverses hexagonal lattice with direction arrows, count paths
Cell phone plan cost with texts and overage minutes
Minimum small bottles to fill a large bottle
Evaluate nested custom average operations
Difference of two arithmetic sequence sums X and Y
Weighted average of running times with grade ratios
Smallest possible size of union of two sets
Identify the equation with no solution
Percentage of geese among non-swans birds
Area of rectangle bounded by given line equations
Cost of pencil in cents given total cost and constraints
Ratio of areas of inscribed square to outer square
Number of free throws given basketball scoring conditions
Count even three-digit numbers with distinct given digits
Probability circle area < circumference from dice sum
Total trip miles given battery and gasoline usage rates
Simplify sum of nested square roots with radicals
Find A+H in sequence with constant sum of consecutive terms
Area inside one circle but outside two others in a configuration
Percent population growth over 20 years given perfect square conditions
Probability that two clockwise chords of length r on a circle intersect
Probability all four selected coins are genuine given equal weight pairs
Number of colorings of pentagon vertices with diagonal color restrictions
Find the last number said after sequential skipping patterns
Volume of intersection of two regular tetrahedra inside a unit cube
Count points dividing unit square into 100 equal-area triangles but not 60
Simplify the difference of two fractions: (2+4+6)/(1+3+5) minus its reciprocal.
Find minimum next test score to raise average by 3 points from given scores.
Rectangular tile dimensions reported to nearest inch, find minimum possible area.
Two people share trip costs; how much must LeRoy give Bernardo to equalize?
Reversed digits of two-digit a gave product 161; find correct product of a and b.
Casper eats fractions of candies and gives some away over three days; find starting amount.
Triangle's two angles sum to 108° and differ by 30°; find the largest angle.
Beach crowded if at least 80°F and sunny; it's not crowded; what can be concluded?
Right triangle ABC, area of triangle EBD is one third of ABC; find BD.
Set of powers of 10 up to 10^10; ratio of largest to sum of others closest to which integer?
52 people, 12 months; largest n such that at least n have same birth month always true.
Track with straight sides and semicircular ends, width 6 m, 36 s longer outside; find speed.
Two real numbers chosen from [-20,10]; probability their product is positive.
Rectangle with diagonal 25 and area 168; find its perimeter.
Custom operation a@b = (a+b)/2; which distributive laws hold for all numbers?
Dart board regular octagon; probability dart lands in center square.
Circle with diameter EB parallel to DC; given angle ratio, find angle BCD.
Rectangle ABCD, AB=6, BC=3, M on AB with ∠AMD = ∠CMD; find ∠AMD.
Solve √(5|x|+8) = √(x^2-16) and find product of all real roots.
Rhombus side 2, ∠B=120°; region R closer to B than other vertices; find area.
Four integers w>x>y>z sum to 44, six pairwise positive differences given; sum of possible w.
Pyramid square base side 1, lateral faces equilateral; inscribed cube; find its volume.
Find the hundreds digit of 2011^2011.
Line y=mx+2 passes through no lattice point for 0<x≤100; max a where 1/2<m<a.
Sequence of triangles from incircle tangency points; perimeter of last valid triangle.
Find average width of five books
Four squares and a rectangle form a large square; find length-to-width ratio of rectangle
Tyrone gives marbles to Eric so he has twice as many; find marbles given
Book recorded onto equal-length discs; find minutes per disc using minimum discs
Circle circumference 24π; find area coefficient k
Evaluate custom operation ♠(2, ♠(2,2)) defined as x - 1/y
Crystal runs north, northeast, southeast, then straight back; find return distance
Tony’s pay based on age; worked 50 days earned $630; find his age at end
Three-digit palindrome x where x+32 is a four-digit palindrome; sum of digits
Marvin's birthday Tuesday May 27, 2008; next year it falls on Saturday
Inequality a ≤ 2x+3 ≤ b has solution interval length 10; find b-a
Scale model water tower; find model height given volume ratio
Angelina drove at two speeds with a gas stop; identify correct distance equation
Triangle with AB=2AC, equilateral △CFE, angle condition; find ∠ACB
Four amphibians (toads always true, frogs always false) make statements; find number of frogs
Triangle ABC, angle bisector BD, AD=3, DC=8, integer sides; smallest perimeter
3x3x3 cube with 2x2 square holes through each face; find remaining volume
Bernardo and Silvia pick three distinct digits and form descending numbers; probability Bernardo's is larger
Equiangular hexagon with side lengths 1 and r; area of △ACE is 70% of hexagon; sum of possible r
Fly visits all vertices of a unit cube, starting and ending same, each once; maximize path length
Cubic x³ - a x² + b x - 2010 with three positive integer roots; smallest possible a
Eight points on a circle, all chords drawn; count interior triangles with vertices inside circle
Boxes with red and white marbles; draw until red; find n where probability of stopping after n draws < 1/2010
Last two non-zero digits of 90!
Jim subtracts largest possible square repeatedly to reach zero; smallest n giving sequence length 8
Evaluate the arithmetic expression 100(100-3) - (100*100 - 3).
Find the percent of a 9-hour work day spent in two meetings totaling 135 minutes.
Minimum socks to pull to guarantee a matching pair with at least 2 of each of 4 colors.
Define a custom average operation ♡(x); find the sum of values at 1, 2, and 3.
A 31-day month has equal Mondays and Wednesdays; how many possible first days?
Given a circle with diameter AB, point C, and ∠COB=50°, find ∠CAB.
Isosceles triangle sides 10,10,12; rectangle width 4 with same area; find its perimeter.
Ticket costs x dollars; groups pay $48 and $64. How many possible whole-number values for x?
Larry ignores parentheses but gets the correct result; find the substituted number e.
Shelby drives 16 miles in 40 minutes at two speeds; find minutes driven in the rain.
Compare coupon discounts; find difference between max and min prices where A saves at least as much.
Survey Yes/No change from 50% to 70%; max minus min possible different answers.
Find the sum of all solutions to the absolute value equation x = |2x - |60-2x||.
The average of 1 through 99 and x is 100x; find x.
Multiple-choice scoring: 4 for correct, -1 for wrong. Jesse scores 99; max number correct.
Square of side 1 and circle of radius √3/3 share center; find area inside circle but outside square.
Andrea's median score among all; teammates placed 37th and 64th; find number of schools.
Random a,b,c from 1–2010; probability that abc+ab+a is divisible by 3.
Circle area 156π, equilateral triangle ABC chord, OA=4√3; find side length of triangle.
Two circles tangent to regular hexagon and lines; find ratio of their areas.
Pick a random 4-digit palindrome; find the probability it is divisible by 7.
Distribute 7 distinct candies into 3 bags, red and blue each receive at least one piece.
Fill a 3x3 grid with digits 1-9 so rows and columns increase; count arrangements.
Basketball quarter scores form increasing geometric and arithmetic sequences; first-half total points.
Polynomial with integer coefficients has P(odd)=a, P(even)=-a; smallest possible a>0.
Find minimum number of 12 oz cans needed to make 128 oz.
Determine which total value cannot be made from four coins (pennies, nickels, dimes, quarters).
Simplify the nested fraction 1 + 1/(1 + 1/(1+1)).
Find required average bike speed for a triathlon given swim and run times.
Find the sum of the digits of 111,111,111 squared.
Find the fraction of the semicircle's area that is shaded, given an inscribed circle.
Find whole milk fat percentage given 2% fat is 40% less than whole milk.
Find total cost for family with children, adults, seniors given senior ticket price.
Find a in geometric sequence a, b, 2009 with integer ratio.
Find area of right triangle given segments on hypotenuse from altitude.
Find volume of cube given volume change when dimensions are shifted by ±1.
Find integer diagonal BD in quadrilateral with given sides using triangle inequality.
Express 12^{mn} in terms of P=2^m and Q=3^n.
Find ratio of longer to shorter side of rectangles forming a square pattern.
Find number of diamonds in the 20th figure of a growing diamond pattern.
Find sum of all possible values of |a-d| given absolute differences of a,b,c,d.
Find length EF where EF is perpendicular to diagonal DB in rectangle ABCD.
Find percentage of non-swimmers who play soccer given overlapping sport percentages.
Find number of possible integer radii r<100 for rolling circle to align.
Find total time until Lauren reaches Andrea after Andrea stops with flat tire.
Ratio of total area of four small circles to area of large enclosing circle.
Find probability sum is 7 after randomly reassigning numbers on two dice.
Find AE in quadrilateral with equal areas of two triangles formed by diagonals.
Probability that three random vertices of a cube determine a plane cutting through interior.
Find maximum number of factors of 2 in 10...064 with k zeros.
Jane buys 50-cent muffin or 75-cent bagel for 5 days; total cost whole dollars. How many bagels?
Simplify the complex fraction ((1/3)-(1/4))/((1/2)-(1/3)).
Painter had paint for 30 rooms, lost 3 cans, now only 25 rooms. Cans used for 25 rooms?
Rectangular yard with two congruent isosceles right triangle flower beds; find fraction of yard occupied.
Twenty percent less than 60 is one-third more than what number?
Product of ages of Kiana and her two older twin brothers is 128; find sum of their ages.
Insert parentheses in 2×3 + 4×5 to get different values; how many distinct values?
Gasoline price rises 20%, falls 20%, rises 25%, falls x%, returns to original; find x to nearest integer.
Segments BD and AE intersect at C; AB=BC=CD=CE; ∠A = 5/2 ∠B. Find ∠D.
Flagpole 5 m tall snaps x m above ground; top touches ground 1 m from base. Find x.
How many 7-digit palindromes using digits 2,2,3,3,5,5,5?
Points A,B,C,D on one line, E,F on parallel line; max distinct areas of triangles from 6 points?
Convex pentagon rolled along x-axis; which side touches x=2009?
Millet seeds: add quart daily, birds eat 25% of millet; when does millet exceed half?
Bucket 2/3 full weighs a, 1/2 full weighs b; find full weight in terms of a and b.
Circle, equilateral triangle ABC with BA, BC tangent; circle meets BO at D; find BD/BO.
Five unit squares; line from (c,0) to (3,3) divides region into equal areas. Find c.
Rectangle 8×6, M midpoint of AC, E on AB with ME ⟂ AC; find area of triangle AME.
Clock mistakenly shows 9 instead of 1; what fraction of day does it show correct time?
Right triangle ABC with AB=1, BC=2; angle bisector from A meets BC at D. Find BD.
Remainder when 3^0+3^1+⋯+3^{2009} is divided by 8.
Cube cake with edge 2, iced on sides and top, cut vertically; find c+s for piece B.
Runners on circular track, laps 90s and 80s; picture taken randomly 10-11 min. Probability both in picture?
Keystone arch of 9 congruent isosceles trapezoids; find larger interior angle of trapezoid.
Cube with random stripe on each face; probability of a continuous stripe encircling the cube?
Doughnut machine starts 8:30 AM, finishes 1/3 at 11:10; find completion time.
Square inside rectangle, width:side=2:1, length:width=2:1; find percent of rectangle area inside square.
Function <n> is sum of proper divisors; evaluate <<<6>>>.
If 2/3 of 10 bananas = 8 oranges, find oranges equal to 1/2 of 5 bananas.
Compute product 8/4 * 12/8 * ... * 2008/2004.
Find average speed for triathlon with equal-distance legs at 3, 20, 10 km/h.
Simplify (3^2008^2 - 3^2006^2)/(3^2007^2 - 3^2005^2).
Heather compares 15% off + $90 rebate vs 25% off, saves $15; find sticker price.
For (2x/3 - x/6) integer, which statement must be true about x?
Square S1 area 16, bisect sides to form smaller squares S2, S3; find S3 area.
Boat 1 mile from shore, water enters 10 gal/min, sinks >30 gal; row 4 mph; find slowest bailing rate.
Red, blue, green marbles with percentage increases; express total in terms of r reds.
Doug paints room in 5h, Dave in 7h, they work together with 1h break; find equation for total time t.
Old TV 4:3 ratio, 27-inch diagonal; movie 2:1 ratio letterboxed; find height of each darkened strip.
Han drove 1h longer, 5mph faster than Ian, extra 70 mi; Jan 2h longer, 10mph faster; find Jan's extra miles.
Two circles, one internally tangent and tangent to 60° angle rays; find ratio of areas.
Equilateral triangle side 6; find area of region outside triangle but within 3 units of its points.
Right triangle perimeter 32 and area 20; find the hypotenuse length.
Rectangle 2x6 rotated 90° twice; find length of path traveled by designated point.
Trapezoid bases 9 and 12, diagonals intersect at K; area of triangle AKD = 24; find trapezoid area.
Cube side 1; plane through two opposite vertices and midpoints of opposite edges; find quadrilateral area.
Sequence defined by coin flips (double/halve subtract 1); probability fourth term is integer.
Choose two subsets of {a,b,c,d,e} with union S and intersection exactly two; count ways (order irrelevant).
Let k=2008^2+2^2008; find the units digit of k^2+2^k.
Six rectangular place mats (width 1, length x) around round table radius 4; find x.
Find number of possible total points from 5 baskets worth 2 or 3 points.
Calculate diagonal sum difference after row reversals in a 4x4 calendar.
Simplify cube root of x times square root of x.
Maximize a single player's salary given team total cap and minimum salary.
Evaluate (x-y)^2 * (y-x)^2 under custom operation a*b=(a-b)^2.
Find what fraction BC is of AD given AB=4BD and AC=9CD on collinear points.
How many unit equilateral triangles fill a large equilateral triangle of side 10?
Count the number of bouquets with roses and carnations costing exactly $50.
Find the average of two real solutions to ax^2 - 2ax + b = 0.
Find length AC where C is the midpoint of minor arc AB on a circle.
Find u5 in a sequence satisfying u_{n+2}=2u_{n+1}+u_n given u3 and u6.
Convert pedometer readings and flips into miles walked by Postman Pete.
Find the 2008th term of a sequence whose mean of first n terms is n.
Rotate point A in a 30-60-90 triangle 90° counterclockwise about origin and find coordinates.
Count integer right triangles with legs a,b, hypotenuse b+1, and b<100.
Probability sum of rolled dice is odd when coins determine number of dice rolled.
Probability exactly one of three randomly selected voters approves of mayor given 70% approval.
Find total bricks in a chimney given individual and combined work rates and output decrease.
Volume of water in a horizontal cylindrical tank with given radius and water depth.
Probability sum of two nonstandard dice is 5, 7, or 9.
Count seating arrangements of 5 married couples alternating gender with spouse restrictions.
Probability that 3 red, 2 white, 1 blue beads in a line have no adjacent same color.
How many integer dimensions a<b for a rectangle with 1-foot border covering half area?
Find angle BAD in quadrilateral with AB=BC=CD, angles 70° and 170°.
How many times do Michael and a garbage truck meet on a straight path with stops?
Susan buys 4 tickets with 25% discount, Pam buys 5 with 30% discount; how much more does Pam pay?
Custom operations a@b and a#b defined; evaluate (6@2)/(6#2).
Aquarium partially filled; brick placed inside; find water rise in cm.
Larger of two consecutive odd integers is 3 times smaller; find sum.
Pencils and notebooks costs given; find total cost of 16 pencils and 10 notebooks.
AMC 10 participation numbers 2002-2007; find largest percentage increase between consecutive years.
Inheritance taxed 20% then 10% on remainder with total tax $10500; find inheritance.
Isosceles triangles ABC and ADC share side AC; given two angles, find angle BAD.
Real numbers a,b satisfy 3^a = 81^{b+2} and 125^b = 5^{a-3}; find ab.
Family average age 20, father 48, mother+children average 16; find number of children.
Numbers 1-8 placed on cube vertices, face sums equal; find common sum.
Two tour guides each must lead at least one of six tourists; number of possible groupings.
Carville to Nikpath takes 4.5 hr at 70 mph; find travel time at 60 mph.
3-4-5 triangle inscribed in circle of radius 3; find area.
Four small circles tangent to square and a central larger circle; find square area.
Integers a,b,c,d from 0 to 2007; probability ad-bc is even.
Positive integers m,n with 75m = n^3; minimize m+n.
12-sided orthogonal polygon ABCDEFGHIJKL; find area of quadrilateral ABCM.
Square painted by sweeping brush along both diagonals; half area painted; find side to width ratio.
Given a + a^{-1} = 4, find a^4 + a^{-4}.
Outer cube surface area 24, sphere inscribed, then inner cube inscribed in sphere; find inner cube surface area.
Sequence of three-digit numbers where tens/units become hundreds/tens of next term; largest prime dividing sum S.
Positive integers m>=n with m^2 - n^2 = 96; count ordered pairs.
Two circles radius 2 centered at A,B; tangents from midpoint O and common tangent EF form shaded region; find its area.
Count positive integers n with n + S(n) + S(S(n)) = 2007 where S(n) is digit sum.
Isabella paints walls of 3 bedrooms each 12x10x8 feet with 60 sq ft openings; find total painted area.
Evaluate (3★5)-(5★3) where a★b=(a+b)b.
Student drives 120 mi at 30 mpg then returns at 20 mpg; find overall average gas mileage.
Given central angles BOC=120°, AOB=140° in circumcircle of triangle ABC; find angle ABC.
Given 'all Arogs are Brafs, all Crups are Brafs, all Dramps are Arogs, all Crups are Dramps'; which statement is implied?
AMC 10 scoring: correct 6 pts, unanswered 1.5 pts. Sarah attempts 22, leaves 3 unanswered; how many correct to score at least 100?
Convex pentagon with equal sides, angles A and B each 90°; find angle E.
Digits of form bbcac with 0≤a<b<c≤9 and b average of a and c; how many such five-digit numbers?
Code shifts each occurrence of a letter right by triangular number; find letter that replaces 12th 's'.
Points B, C fixed; set S of points A such that triangle ABC has area 1; describe S.
Circle through vertices of isosceles triangle sides 3,3, base 2; find area of circle.
Tom's age T equals sum of his three children's ages; N years ago his age was twice sum of children's ages; find T/N.
Two circles radius 2 centered at (2,0) and (0,2); find area of intersection.
Group initially 40% girls; 2 girls leave, 2 boys arrive, then 30% girls; find initial number of girls.
Quadrilateral angles satisfy A=2B=3C=4D; find angle A rounded to nearest degree.
Class 10% juniors, 90% seniors; overall average 84, senior average 83; find junior score.
Point P inside equilateral triangle ABC; distances from P to sides 1,2,3; find side length AB.
Circle radius 1 surrounded by 4 equal circles of radius r; find r as shown.
Wheel spun twice, remainders mod 4 and 5 designate checkerboard square; find probability shaded square.
5x5 grid of blocks; number of ways to choose 3 blocks with no two same row or column.
Square inscribed in right triangle 3-4-5 with two vertices on hypotenuse; find side length.
Choose 1-4 then roll two four-sided dice; win $1 if one shows number, $2 if both, lose $1 if none; find expected return.
Pyramid with square base cut by plane 2 units above base; smaller top pyramid surface area half original; find altitude.
Smallest positive integer divisible by 4 and 9 using only digits 4 and 9 (at least one each); find its last four digits.
Number of coprime positive integer pairs (a,b) such that a/b + 14b/(9a) is integer.
Cost to purchase 5 sandwiches at $3 each and 8 sodas at $2 each.
Evaluate h⊗(h⊗h) given the operation x⊗y = x³ - y.
Mary's age to Alice's age ratio 3:5; Alice is 30, find Mary's age.
Find the largest possible sum of digits on a digital watch (hours, minutes, AM/PM).
Pizza with 8 slices, plain $8, anchovy half $2 extra. Dave ate anchovy slices+1 plain. Find extra Dave paid.
Solve (7x)¹⁴ = (14x)⁷ for non-zero real x.
Rectangle 8×18 cut into two congruent hexagons; rearranged to square. Find segment y.
Find c for parabola y=x²+bx+c passing through (2,3) and (4,3).
How many sets of ≥2 consecutive positive integers sum to 15?
How many real x make √(120-√x) an integer?
Describe the graph of (x+y)² = x² + y².
Dog tied with 8-ft rope to a 16-ft square shed; compare roaming areas of two arrangements.
Fair game: pay $5; roll die twice: win if both even and match. Find winning prize.
Rings of thickness 1 cm, diameters decreasing from 20 to 3 cm. Find total height.
Two runners on concentric circular tracks, opposite directions, different radii and speeds. Number of meetings in 30 min.
Two tangent circles of radii 1 and 2, isosceles triangle tangent to both circles. Find triangle area.
In rectangle, trisected sides, find area of central quadrilateral formed by diagonals.
Count distinct license plates with 4 digits and 2 adjacent letters (any order, repeating allowed).
Count non-similar triangles with angles as distinct positive integers in arithmetic progression.
Probability that among six distinct integers from 1-2006, some pair's difference is a multiple of 5.
Count 4-digit integers containing at least one digit 2 or 3.
Pigs $300, goats $210; find smallest positive debt expressible as 300p+210g.
Two circles radii 3 and 8 with internal tangent CD, AB meets CD at E, AE=5. Find CD.
Find volume of regular octahedron formed by joining centers of adjacent faces of unit cube.
Prob bug visits all 8 vertices exactly once in 7 random edge moves on a cube.
Sum of alternating powers of -1 from 1 to 2006
Custom operation (x+y)(x-y); compute 3 ♠ (4 ♠ 5)
Total points 34, win margin 14; find Panthers' score
Concentric circles diameter 1 and 3; ratio of ring area to inner circle
Pack 2x3 and 3x4 rectangles in smallest square without overlap
Semicircular arcs on square side 2/π; find perimeter of region
Simplify sqrt(x/(1-(x-1)/x)) for x<0
Square area 40 inscribed in semicircle; find area of semicircle
Lemonade with 100g lemon, 100g sugar, 400g water; calories in 200g
Triangle sides: one 3x another, third 15; greatest integer perimeter
Tens digit of sum 7!+8!+...+2006!
Lines x=y/4+a, y=x/4+b intersect at (1,2); find a+b
Joe drinks 2 oz then adds cream; JoAnn adds cream then drinks 2 oz; cream ratio
Roots a,b of x^2-mx+2=0; new roots a+1/b, b+1/a; find product q
Similar rhombi ABCD and BFDE, angle BAD=60°, area ABCD=24; find area BFDE
Feb 29, 2004 is Sunday; day of week for Feb 29, 2020
Bag swap: Alice moves one ball, Bob moves one; probability bags identical
Recursive sequence a_n=a_{n-1}/a_{n-2}, a1=2,a2=3; find a2006
Circle radius 2, square OABC side 1; find shaded area bounded by BD, BE, arc DE
Rectangle ABCD with A(6,-22), B(2006,178), D(8,y); find area
Dice probabilities in ratio 1:2:3:4:5:6; probability of sum 7
Sandwiches cost 4¢B+5¢J per, total $2.53; find cost of jam
Triangle partitioned into 3 triangles and quadrilateral; areas 3,7,7; find shaded area
Externally tangent circles radii 2 and 4; common tangents; area of hexagon AOBCPD
8 children ages include 9; 4-digit number divisible by all; which age not possible?
Mike and Joe each tip $2, at 10% and 20% of their bills. Find bill difference.
Custom operation (a★b)=(a+b)/(a-b). Compute ((1★2)★3).
Equations 2x+7=3 and bx-10=-2 share same solution. Find b.
Rectangle diagonal x, length twice width. Express area in terms of x.
Store offers one free per four purchased. Dave needs 7, Doug 8. Savings together vs. separately.
Average of 20 numbers is 30, average of another 30 is 20. Find overall average of 50 numbers.
Josh rides twice as long as Mike at four-fifths his speed. They meet after 13 miles total. Find Mike's distance.
Square with side sqrt(50), inscribed inner square with BE=1. Find area of inner square.
Three X and two O tiles randomly arranged. Probability of reading XOXOX.
Quadratic 4x^2+ax+8x+9=0 has one solution. Find sum of possible a.
Cube painted red, cut into unit cubes; exactly 1/4 of all faces red. Find n.
Trefoil from circular sectors on equilateral triangles, base length 2. Find area.
Inequality (130n)^50 > n^100 > 2^200. Count positive integer solutions n.
Three-digit numbers where middle digit is average of first and last digits. How many?
How many positive cubes divide 3! * 5! * 7! exactly?
Two-digit number minus sum of digits yields unit digit 6. Count possibilities.
Five-pointed star with vertices 3,5,6,7,9; edge sums form arithmetic progression. Find middle sum.
Best-of-5 series; A wins series, B wins game 2. Probability B won game 1.
Rotated unit square lowered between two fixed unit squares. Find distance of lifted vertex from base.
Equiangular octagon with alternating sides 1 and sqrt(2)/2. Find area.
Sum 1+2+...+n evenly divides 6n. Count positive integers n.
S is first 2005 multiples of 4, T first 2005 multiples of 6. How many common elements?
Diameter AB with point C (2*AC=BC), DC perp AB, DE diameter. Ratio of areas of triangles DCE and ABD.
Both P(n)=sqrt(n) and P(n+48)=sqrt(n+48) hold (P is greatest prime factor). How many n?
Triangle ABC with side lengths 25,39,42. Points D and E on AB, AC divide. Ratio of area ADE to quadrilateral BCED.
Scout troop buys 1000 candy bars at 5 for $2, sells at 2 for $1; find profit.
Find positive number x such that x% of x equals 4.
Gallon of paint; 1/3 used first day, 1/3 of remainder second; fraction left for third day.
Evaluate (5⋄12)⋄((-12)⋄(-5)) where a⋄b = √(a²+b²).
Brianna spends 1/5 of money to buy 1/3 of CDs; what fraction of money remains after buying all?
Lisa aims for A on 80% of 50 quizzes; has 22 A's after 30; how many more non-A's can she afford?
Circle inscribed in square, then square inscribed in that circle, then circle inscribed; find area ratio smallest circle to largest square.
An 8×10 ft floor tiled with 1-ft tiles, each with four quarter-circles; find total shaded area.
Two dice with faces 1,1,2,2,3,3 and 4,4,5,5,6,6; find probability sum is odd.
Isosceles triangle legs 7, base 2; point D on extended base with CD=8; find BD.
Sequence starts 2005, each term is sum of cubes of digits; find 2005th term.
Twelve fair dice rolled; find probability product of numbers is prime.
Count numbers 1–2005 that are multiples of 3 or 4 but not 12.
Equilateral triangle side 2, M midpoint of AC, C midpoint of BD; find area of triangle CDM.
Eight bills: 2 each of $1, $5, $10, $20; draw two; find probability sum ≥ $20.
Quadratic x²+mx+n has roots twice those of x²+px+m; find n/p.
If 4^a=5, 5^b=6, 6^c=7, 7^d=8, find a·b·c·d.
Phone numbers 555-abc-defg, digits a–g distinct, increasing, none 0 or 1; how many possible numbers?
Score distribution: 10% 70, 25% 80, 20% 85, 15% 90, rest 95; find mean minus median.
Average of all 5-digit numbers using digits 1,3,5,7,8 exactly once each.
Forty slips, numbers 1–10 each four times; draw four; find ratio q/p where p = all same number, q = two of one and two of another.
For how many n ≤ 24 is n! divisible by 1+2+...+n?
Trapezoid ABCD, AB||DC, midpoints E,F of BC,DA; area ABEF twice area FECD; find AB/DC.
Two-digit x and reversed y satisfy x²−y²=m²; find x+y+m.
Maximum size of subset of {1,…,100} with no two elements summing to 125.
Six people split $1500 equally; find each person's amount.
Given custom operation ⊗(a,b,c)=a/(b-c), evaluate a nested expression.
Alicia earns $20/hr with 1.45% tax; find tax in cents per hour.
Solve |x-1| = |x-2| for x.
Probability that three random points from a 3x3 grid are collinear.
Bertha's daughters and granddaughters total 30; find how many have no daughters.
Oranges stacked in a pyramid with 5x8 base; each layer reduces; find total oranges.
Token game: players start 15,14,13; each round leader gives away tokens; how many rounds?
Overlapping right triangles; find difference of areas of two smaller triangles.
Probability that number of heads is equal when coin A flipped 3 times and B 4 times.
Jar diameter increased 25% while volume unchanged; find percent height decrease.
Hamburger: 3 meat choices and any subset of 8 condiments; total number of kinds.
12 men each dance with 3 women, each woman with 2 men; find number of women.
Average coin value 20 cents; adding one quarter makes average 21; how many dimes?
Given -4 ≤ x ≤ -2, 2 ≤ y ≤ 4, maximize (x+y)/x.
5x5 grid of squares; count squares that contain the center black square.
Two runners start diametrically opposite on circular track; find track length from meeting points.
Arithmetic progression 9, ...; modified to geometric progression; smallest third term?
Cylinder stripe width 3 makes two revolutions; find stripe area.
Square with equilateral triangle inside; ratio of areas of two triangles.
Two lines through center of three concentric circles; shaded area 8/13 of unshaded; find angle.
Square side 2, tangent from vertex to semicircle meets side; find segment length.
Three circles externally tangent, internally tangent to larger circle; find radius of smaller.
Function defined by f(1)=1, f(2n)=n·f(n); evaluate f(2^100).
Three unit spheres on plane support sphere radius 2; distance from plane to top.
Find the number of seats in rows 12 through 22 of an amphitheater with 33 seats per row.
How many two-digit positive integers have at least one digit as 7?
Jenny doubles her free throws each practice; made 48 at fifth practice; find her first practice amount.
Roll a six-sided die; product P of the five visible numbers; find largest certain divisor of P.
Maximize c·a^b - d using the digits 0,1,2,3 each exactly once.
Which product of two factorials among the choices is a perfect square?
Isabella exchanges US dollars for Canadian dollars at 7:10 rate, spends 60 CAD, ends with d CAD; find digit sum of d.
Airport is 8 miles SW of St. Paul and 10 miles SE of Minneapolis; find the distance between the two downtowns.
Square side 10 and circle radius 10 centered at one vertex; find area of their union.
Display of cans: top row 1, each lower row has 2 more than row above; total 100 cans; how many rows?
Two eight-sided dice rolled; find probability that product of the two top numbers exceeds their sum.
Annulus between radii b and c, with tangent segment a; express its area in terms of a, b, c, d, or e.
Stack of US coins exactly 14 mm tall with given thicknesses; how many coins are in the stack?
Bag of red and blue marbles; add red, then yellow, then double blue; find final fraction blue.
Patty has 20 nickels and dimes; if swapped, value would be 70 cents more; find her coins' total worth.
Three unit circles externally tangent to each other and internally tangent to a larger circle; find large radius.
Jack's age digits are Bill's reversed; in 5 years Jack will be twice Bill; find their current age difference.
Right triangle ACE, points B, D, F on sides with given lengths; find area ratio of triangle DBF to ACE.
Sequence defined by a_n = a_{n-2}+a_{n-1}-a_{n-3}; find the 2004th term given first three terms 2001,2002,2003.
Triangle with cevians AD and BE intersecting at T; AT/DT=3, BT/ET=4; find CD/BD.
Union of first 2004 terms of two arithmetic progressions: 1,4,7,... and 9,16,23,...; how many distinct numbers?
Right triangle with sides 5,12,13; find distance between the centers of its inscribed and circumscribed circles.
Each face of a cube painted red/blue with equal independent probability; probability that four vertical faces can be made all same color when placed on a surface.
Triangle ABC with sides 7,8,9; D on circumcircle so AD bisects angle A; find AD/CD.
Circle radius 1 internally tangent to two circles radius 2 at ends of its diameter; find shaded area outside small circle but inside the large circles.
Difference between sum of first 2003 even and sum of first 2003 odd counting numbers
Socks $4, T-shirt $5 more, each member needs 2 socks and 2 shirts, total $2366; find members
15x10x8 cm box, 3 cm cube removed from each corner; percent volume removed
Anna walks 1 km uphill 30 min, downhill 10 min; average speed for round trip
Roots d,e of 2x^2+3x-5=0; find (d-1)(e-1)
Define x ♡ y = |x-y|; which statement is not true?
How many noncongruent triangles with perimeter 7 have integer side lengths?
Probability randomly chosen positive factor of 60 is less than 7
Simplify nested radicals: ∛(x∛(x∛(x√x)))
Attach one square to 4-square polygon at 9 positions; how many fold to cube missing one face?
AMC10 + AMC12 = 123422; find A+M+C
Random point in rectangle (0,0)-(4,1); probability x<y
Three numbers sum to 20, first 4× sum of others, second 7× third; find product
Largest integer product of distinct primes d, e, 10d+e with single-digit d,e; sum of its digits
Probability integer 1..100 divisible by 2 and not by 3
Units digit of 13^2003
Perimeter of equilateral triangle equals area of its circumscribed circle; find radius
Sum of reciprocals of roots of (2003/2004)x+1+1/x=0
Area of lune between semicircle diameter 1 atop semicircle diameter 2
Probability base-10 3-digit number also 3-digit in base 9 and base 11
Select 6 cookies from 3 types, at least 6 of each; how many assortments?
Rectangle ABCD, BH=6, DE=4, find GF where lines intersect
Toothpick equilateral triangle with base row of 2003 small triangles; total toothpicks
Stack 5 red 1-5 and 4 blue 3-6 alternating, red divides neighboring blue; sum middle three
n is 5-digit, q quotient, r remainder when divided by 100; how many n with q+r divisible by 11?
Simplify the fraction (2-4+6-8+10-12+14)/(3-6+9-12+15-18+21).
Al takes one green and one pink pill daily for two weeks; total cost $546, green costs $1 more.
Sum of 5 consecutive even integers is 4 less than sum of first 8 odd numbers. Find smallest even.
Rectangular flower bed with five regions; minimize planting cost by assigning flower types to areas.
Moe mows a 90x150 ft lawn with 28-inch swath overlapping 4 inches, speed 5000 ft/hr. Time?
TV screen diagonal 27 inches, aspect ratio 4:3. Find horizontal length.
Compute sum of floor(sqrt(n)) for n from 1 to 16.
Geometric sequence: second term 2, fourth term 6. Find possible first term.
Solve 25^{-2} = 5^{48/x} / (5^{26/x} * 25^{17/x}) for x.
Old license plates: 1 letter + 4 digits. New: 3 letters + 3 digits. Find increase factor.
Lines with slopes 3 and 5 intersect at (10,15). Find distance between their x-intercepts.
Al, Betty, Clare split $1000. After one year, Al loses $100, others double. Total $1500. Find Al's original.
For two-digit x, digit sum function ♣(x). Find number of x with ♣(♣(x))=3.
Express 3^8 * 5^2 as a^b with positive integers a,b; minimize a+b.
100-player single elimination tennis tournament, 28 byes. Total matches played?
Restaurant: 3 desserts, appetizers = 2 * main courses. Dinners must cover 365 days. Minimum main courses?
Ice cream cone: sphere of ice cream melts to fill cone at 75% volume. Ratio height:radius?
Largest integer dividing (n+1)(n+3)(n+5)(n+7)(n+9) for all positive even n.
Three unit semicircles on diameter of larger semicircle radius 2. Shaded area within large semicircle.
Rectangle ABCD, points F,G on CD, lines AF and BG intersect at E. Find area of triangle AEB.
Bag: 2 red, 2 green beads. Draw and replace with red each time. Probability all red after 3 replacements.
Clock chimes once at half-hour, hour number on hour. Starting 11:15 AM Feb 26, 2003. Date of 2003rd chime?
Regular octagon area 1. Find area of rectangle formed by two opposite sides.
Arithmetic sequence: first four terms are x+y, x-y, xy, x/y. Find fifth term.
How many 4-digit numbers divisible by 3 and ending in 23?
Find the value closest to a fraction involving powers of 10.
Evaluate a custom-defined three-variable operation for given numbers.
Count distinct values from parenthesizing 2^2^2^2 differently.
Find how many positive integers m satisfy m*n ≤ m+n for some n.
Area of shaded region between concentric circles and smaller tangent circles.
Correct an arithmetic mistake by working backwards.
Ratio of areas of two circles given equal arc lengths.
Compare areas of blue triangles, white squares, and the red square in a flag design.
Find the average of A, B, C from two linear equations with large coefficients.
Sum of all roots of a factored quadratic equation.
Minimum number of 1.44 MB disks to store files of given sizes.
Find speed needed to arrive exactly on time given late/early trips.
Shortest altitude of a triangle with sides 15, 20, 25.
Number of possible k such that x^2 - 63x + k = 0 has prime roots.
Sum of four two-digit primes formed using given digits once each.
Find a+b+c+d from a chain of equal expressions.
Fraction of cream in cup 1 after pouring coffee and cream back and forth.
Smallest sum of visible pips on the surface of a 3x3x3 cube of dice.
Area accessible to a dog tethered outside a regular hexagon doghouse.
Ratio HC/JE given parallel segments and equidistant points on line AF.
Largest possible integer in an 8-integer set with mean, median, mode, range all 8.
Number of operations to reduce 100 tiles to 1 by removing perfect squares.
Find AB such that perimeter of triangle AED is twice that of isosceles BEC.
Probability Sergio's number exceeds the sum of Tina's two distinct numbers.
Area of trapezoid with bases 39, 52 and legs 5, 12.
Simplify the quotient of exponentials: 2^{2001} * 3^{2003} / 6^{2002}.
Evaluate the custom operation D(2,4,6) where D(a,b,c)=abc/(a+b+c).
Find which digit is missing from the 9-digit arithmetic mean of {9,99,...,999999999}.
Simplify (3x-2)(4x+1)-(3x-2)4x+1 and evaluate at x=4.
Two externally tangent circles radii 2 and 3 inside a larger circumscribing circle; find the shaded area between them.
Find how many positive integers n make n^2-3n+2 a prime number.
Given 1/2+1/3+1/7+1/n is an integer, determine which statement about n is false.
If July has five Mondays, which day of the week must occur five times in August?
Find the alphabetical position of USAMO among all 5-letter arrangements of A,M,O,S,U.
Find the pair (a,b) of nonzero real numbers that are roots of x^2+ax+b=0.
Three consecutive positive integers have product 8 times their sum; find the sum of their squares.
Find k for which (x-1)/(x-2) = (x-k)/(x-6) has no solution for x.
Find x such that 8xy-12y+2x-3=0 is true for all y.
If N^2 = 25^64 * 64^25, find the digit sum of N.
Positive integers A,B,A-B,A+B are prime; find the nature of their sum.
Find how many integers n make n/(20-n) a perfect square.
A regular octagon of side length 2; find area of triangle ADG.
Four distinct circles in a plane: maximum number of intersection points of at least two circles.
Arithmetic sequence with sums of first 100 and second 100 terms given; find common difference.
Given a-7b+8c=4 and 8a+4b-c=7, find a^2-b^2+c^2.
Three mowers with different lawn sizes and speeds; who finishes first?
Right triangle with midpoints; legs' segments to opposite vertices given; find hypotenuse.
Sequence defined by a_1=1 and a_{m+n}=a_m+a_n+mn; find a_12.
Ferris wheel radius 20 ft, 1 rev/min; time to go from bottom to 10 ft above bottom.
Adding 15 to a list increases mean by 2; adding 1 then decreases mean by 1; find original number of integers.
Median of nine-number list is 10; find the mean.
x is 2 more than product of its reciprocal and additive inverse; find interval containing x.
Sum of two numbers is S; add 3 to each then double; find new sum.
Maximum number of intersection points of a circle and a triangle.
How many of twelve pentominoes have at least one line of symmetry?
Two-digit N equals sum of product and sum of its digits; find units digit.
Decimal point moved four places right gives four times reciprocal; find original number.
Four tutors work every 3,4,6,7 days; today all work; find next day all work together.
Tax rate p% on first $28000, (p+2)% on excess; total tax (p+0.25)% of income; find income.
Given xy=24, xz=48, yz=72 with positive x,y,z; find x+y+z.
Number of unit squares in the 100th ring around a center square.
Product of three consecutive integers divisible by 7; which choice is not necessarily a divisor?
Phone number digits decreasing in three parts with parity constraints; find A.
140 tickets sold for $2001, some full price, some half price; find amount from full-price tickets.
Street with parallel curbs 40 ft apart; crosswalk stripes 50 ft long, curb segment 15 ft; find stripe distance.
Mean of three numbers is 10 more than least, 15 less than greatest; median is 5; find sum.
Cone formed from 252° sector of circle radius 10; find correct slant height and radius.
Plane tiled by squares and pentagons; find percent enclosed by pentagons.
Number of ways to choose four donuts from three types.
Regular octagon formed by cutting isosceles right triangles from a square side 2000; find side length.
Cylinder with diameter=height inscribed in cone of base diameter 10 and height 12; find cylinder radius.
Magic square with five entries unknown; find y+z.
Draw chips without replacement until all red or all white drawn; probability last chip drawn is white.
Trapezoid with perpendicular bases, AB+CD=BC, AD=7; find product AB·CD.
Positive integers ≤2001 multiples of 3 or 4 but not 5; how many?
Positive distinct integers I, M, O with product 2001; find largest possible sum.
Simplify 2000(2000^2000) using exponent rules.
Jenny ate 20% of jellybeans each day; after 2 days 32 remain. Find original number.
Internet bill: fixed fee plus hourly charge; December $12.48, January $17.54 with twice the hours. Find fixed fee.
Triangle PAB, M,N midpoints; P moves parallel to AB. How many of four quantities change?
Fibonacci sequence; which decimal digit appears last in the units position?
Rectangle with AD=1, DB and DP trisect angle ADC; find perimeter of triangle BDP.
2/5 of freshmen and 4/5 of sophomores took test; counts equal. Which ratio must hold?
Given |x-2|=p with x<2, evaluate x-p.
Triangles with sides 4,6,x and 4,6,y; find smallest positive number not a possible |x-y|.
Two different primes between 4 and 18; product minus sum equals which of given numbers?
Pattern of squares: figure 0 has 1, figure 1 has 5, etc. How many squares in figure 100?
Place 5 yellow, 4 red, 3 green, 2 blue, 1 orange peg on triangular board; no row/column same color. How many ways?
Five exam scores entered randomly; after each entry the average is integer. Find the last score entered.
Non-zero reals a,b satisfy ab = a-b. Find possible value of a/b + b/a - ab.
Lattice points; segment AB meets CD at E. Find AE length.
Coin machine: quarter→5 nickels, nickel→5 pennies, penny→5 quarters. Start with 1¢. Which amount possible?
Walk around a 5km square; see 1km horizontally. Find area seen.
Right triangle, point on hypotenuse; lines parallel to legs form square and two small right triangles. Area ratio given, find other ratio.
Nonnegative integers A,M,C sum to 10. Maximize AMC+AM+MC+CA.
Alligators, ferocious creatures, creepy crawlers logical statements; which must be true?
Family drinks 8oz coffee+milk; Angela drinks 1/4 of total milk, 1/6 of total coffee. Find number of people.
List: 10,2,5,2,4,2,x. Mean, median, mode form non-constant arithmetic progression. Find sum of possible x.
Function with f(x/3)=x^2+x+1. Find sum of z such that f(3z)=7.
Year N: 300th day Tuesday; year N+1: 200th day Tuesday. Find day of week for 100th day of N-1.