AIME Problems

780 problems · 2000–2025

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780 of 780 problems

	Description	Topic
2025 AIME I	Find sum of integer bases b>9 where 17_b divides 97_b.	Number Theory	Modular Arithmetic	1
2025 AIME I	Find area of heptagon formed by reflections in triangle with given side ratios.	Geometry	Triangles & Polygons	2
2025 AIME I	Count assignments of 9 players to 3 ice cream flavors with ordering constraints.	Combinatorics	Counting	3
2025 AIME I	Count integer pairs (x,y) in range satisfying 12x^2 - xy - 6y^2 = 0.	Algebra	Systems & Algebraic Manipulation	4
2025 AIME I	Count 8-digit permutations of 1-8 divisible by 22, subtract 2025.	Combinatorics	Counting	5
2025 AIME I	Find r^2+s^2 for isosceles trapezoid with inscribed circle radius 3 and area 72.	Geometry	Triangles & Polygons	6
2025 AIME I	Probability last word contains G when pairing 12 letters into 6 words.	Combinatorics	Probability & Expected Value	7
2025 AIME I	Find sum of k values for which complex system has exactly one solution.	Algebra	Complex Numbers	8
2025 AIME I	Find y-coordinate of intersection of parabola and its 60-degree rotation.	Algebra	Functions	9
2025 AIME I	Count ways to fill 3x9 Sudoku-like grid with numbers 1-9.	Combinatorics	Counting	10
2025 AIME I	Sum y-coordinates of intersections of sawtooth function and parabola x=34y^2.	Algebra	Functions	11
2025 AIME I	Find area of finite region in plane x+y+z=75 satisfying inequalities.	Algebra	Inequalities & Optimization	12
2025 AIME I	Expected number of regions when 27 segments divide a disk.	Combinatorics	Probability & Expected Value	13
2025 AIME I	Minimize sum of distances from point X to vertices of convex pentagon.	Geometry	Triangles & Polygons	14
2025 AIME I	Count triples (a,b,c) up to 3^6 with a^3+b^3+c^3 divisible by 3^7.	Number Theory	Factors, Multiples & Divisibility	15
2025 AIME II	Find area of triangle BGE given collinear points and distances.	Geometry	Triangles & Polygons	1
2025 AIME II	Sum of positive n such that n+2 divides 3(n+3)(n^2+9).	Number Theory	Modular Arithmetic	2
2025 AIME II	Number of colorings of 2x2 grid edges with 2 red and 2 blue per square.	Combinatorics	Counting	3
2025 AIME II	Evaluate product of logarithms and find sum of numerator and denominator.	Algebra	Logarithms & Exponents	4
2025 AIME II	Find sum of arcs on circumcircle of medial triangle given triangle angles.	Geometry	Triangles & Polygons	5
2025 AIME II	Find area of rectangle inscribed in circle tangent to larger circle.	Geometry	Circular Geometry	6
2025 AIME II	Probability that LCM of random subset of divisors of 2025 is 2025.	Combinatorics	Probability & Expected Value	7
2025 AIME II	Count N up to 1000 where greedy coin algorithm uses fewest coins.	Combinatorics		8
2025 AIME II	Find n+t where n zeros of f(x)=sin(7π sin(5x)) and t tangent zeros.	Algebra	Functions	9
2025 AIME II	Number of subsets of 16 chairs with 8 people, no three adjacent.	Combinatorics	Counting	10
2025 AIME II	Number of ways to draw 12 equal segments pairing vertices of 24-gon.	Combinatorics	Counting	11
2025 AIME II	Find A1A2 + A1A11 for 11-gon with given area and cosine conditions.	Geometry	Trigonometry	12
2025 AIME II	Find m+n mod 1000 for x_2025 = m/n from given recurrence.	Algebra	Sequences & Series	13
2025 AIME II	Find n where area of quadrilateral BKLC equals n√3 in right triangle.	Geometry	Triangles & Polygons	14
2025 AIME II	Sum of k such that f(x) minimum occurs at exactly two positive x.	Algebra	Polynomials	15
2024 AIME I	Find walk time including coffee break at new speed.	Algebra	Systems & Algebraic Manipulation	1
2024 AIME I	Find product xy given two logarithmic equations.	Algebra	Logarithms & Exponents	2
2024 AIME I	Count n ≤ 2024 where Bob wins a token removal game.	Combinatorics	Counting	3
2024 AIME I	Find conditional probability of winning grand prize given a prize.	Combinatorics	Probability & Expected Value	4
2024 AIME I	Find length CE in configuration of two rectangles and a circle.	Geometry	Circular Geometry	5
2024 AIME I	Count grid paths of length 16 with exactly four direction changes.	Combinatorics	Counting	6
2024 AIME I	Maximize real part of complex expression with \|z\|=4.	Algebra	Complex Numbers	7
2024 AIME I	Find inradius of triangle given two sequences of tangent circles.	Geometry	Triangles & Polygons	8
2024 AIME I	Find greatest lower bound for BD^2 in rhombus on hyperbola.	Algebra	Coordinate Geometry	9
2024 AIME I	Find length AP where AD meets circumcircle again.	Geometry	Triangles & Polygons	10
2024 AIME I	Probability octagon can be rotated so blues go to former red positions.	Combinatorics	Counting	11
2024 AIME I	Find number of intersections of two graphs involving absolute values and trig.	Algebra	Functions	12
2024 AIME I	Find smallest m such that p^2 divides m^4+1 for least prime p.	Number Theory	Modular Arithmetic	13
2024 AIME I	Find inradius of tetrahedron with given edge lengths.	Geometry	3D Geometry & Solids	14
2024 AIME I	Find r^2 for smallest sphere containing all boxes with given surface area and volume.	Algebra	Inequalities & Optimization	15
2024 AIME II	Residents own items; find number owning all four.	Combinatorics	Counting	1
2024 AIME II	List of positive integers with given sum, mode, median; find sum of squares.	Combinatorics	Counting	2
2024 AIME II	2x3 grid digits; row sums to 999, column sums to 99.	Combinatorics	Counting	3
2024 AIME II	System of log equations; find absolute value of log expression.	Algebra	Logarithms & Exponents	4
2024 AIME II	Equilateral hexagon with parallel opposite sides; find side length.	Geometry	Triangles & Polygons	5
2024 AIME II	Sets of positive integers with max in A; find sum of A.	Combinatorics	Counting	6
2024 AIME II	Four-digit number; changing any digit to 1 gives multiple of 7.	Number Theory	Modular Arithmetic	7
2024 AIME II	Torus inside and outside sphere; find difference of radii of tangency circles.	Geometry	3D Geometry & Solids	8
2024 AIME II	Place chips in 5x5 grid with row/column color constraints.	Combinatorics	Counting	9
2024 AIME II	Triangle with incenter I, circumcenter O, IA perpendicular OI; find AB*AC.	Geometry	Triangles & Polygons	10
2024 AIME II	Nonnegative integers a,b,c sum to 300; find number of triples satisfying equation.	Algebra	Polynomials	11
2024 AIME II	Family of unit segments in first quadrant; find unique point C on AB.	Geometry	Coordinate Geometry	12
2024 AIME II	Product over 13th roots of unity; find remainder modulo 1000.	Algebra	Complex Numbers	13
2024 AIME II	Base-b two-digit numbers where digit sum equals sqrt of number.	Number Theory	Base Representations	14
2024 AIME II	Rectangles formed by sides/diagonals of a regular dodecagon.	Combinatorics	Counting	15
2023 AIME I	Probability that each man stands opposite a woman in a circle	Combinatorics	Probability & Expected Value	1
2023 AIME I	Solve system of log equations for rational n	Algebra	Logarithms & Exponents	2
2023 AIME I	Count intersection points of 40 lines with given multiplicities	Combinatorics	Counting	3
2023 AIME I	Sum of m such that 13!/m is a perfect square	Number Theory	Factors, Multiples & Divisibility	4
2023 AIME I	Area of square given products of distances from point on circumcircle	Geometry	Triangles & Polygons	5
2023 AIME I	Expected number of correct guesses in a card guessing game	Combinatorics	Probability & Expected Value	6
2023 AIME I	Count integers with distinct remainders modulo 2 through 6	Number Theory	Modular Arithmetic	7
2023 AIME I	Perimeter of rhombus given distances from point on incircle to sides	Geometry	Triangles & Polygons	8
2023 AIME I	Count cubic polynomials with unique integer m not 2 satisfying p(m)=p(2)	Algebra	Polynomials	9
2023 AIME I	Find a and sum of floor terms between -1000 and 1000	Number Theory	Modular Arithmetic	10
2023 AIME I	Number of subsets of {1..10} with exactly one pair of consecutive integers	Combinatorics	Counting	11
2023 AIME I	Find tan^2 of equal angles in equilateral triangle with given side divisions	Geometry	Triangles & Polygons	12
2023 AIME I	Ratio of volumes of two noncongruent parallelepipeds with rhombus faces	Geometry	3D Geometry & Solids	13
2023 AIME I	Number of sequences of 144 hand movements visiting all positions exactly once	Combinatorics	Counting	14
2023 AIME I	Largest prime p<1000 with integer complex z satisfying triangle condition	Algebra	Complex Numbers	15
2023 AIME II	Six apple tree counts form an arithmetic sequence; greatest is double least; sum is 990.	Algebra	Sequences & Series	1
2023 AIME II	Find greatest integer < 1000 that is a palindrome in base 10 and base 8.	Number Theory	Base Representations	2
2023 AIME II	Isosceles right triangle ABC with interior point P; AP=10; find area of triangle.	Geometry	Triangles & Polygons	3
2023 AIME II	Real numbers x,y,z satisfy three symmetric equations; find sum of squares of possible x.	Algebra	Systems & Algebraic Manipulation	4
2023 AIME II	Positive rational r such that r and 55r have same numerator+denominator sum in lowest terms.	Number Theory	Factors, Multiples & Divisibility	5
2023 AIME II	Two points chosen uniformly in L-shaped region of three unit squares; probability midpoint is inside.	Combinatorics	Probability & Expected Value	6
2023 AIME II	Color vertices of regular dodecagon red/blue; no four same-colored vertices form a rectangle.	Combinatorics	Counting	7
2023 AIME II	Product over k=0..6 of (omega^{3k}+omega^k+1) where omega is primitive 7th root of unity.	Algebra	Complex Numbers	8
2023 AIME II	Two intersecting circles with common tangent; parallel line through P; find area of trapezoid XABY.	Geometry	Circular Geometry	9
2023 AIME II	Place numbers 1-12 in 2x6 grid so adjacent cells differ by not divisible by 3.	Combinatorics	Counting	10
2023 AIME II	Number of collections of 16 distinct subsets of {1,2,3,4,5} with pairwise nonempty intersection.	Combinatorics	Counting	11
2023 AIME II	Triangle ABC with sides 13,14,15; M midpoint BC; P on circumcircle; find AQ on AM.	Geometry	Triangles & Polygons	12
2023 AIME II	Acute angle A with tan A = 2 cos A; count n ≤ 1000 where sec^n A + tan^n A ends in 9.	Algebra	Sequences & Series	13
2023 AIME II	Cube with vertices A,B,C,D on plane; heights given; water surface at height 7; find water volume.	Geometry	3D Geometry & Solids	14
2023 AIME II	a_n is least positive multiple of 23 congruent to 1 mod 2^n; count n ≤ 1000 with a_n = a_{n+1}.	Number Theory	Modular Arithmetic	15
2022 AIME I	Find P(0)+Q(0) for two quadratics through (16,54) and (20,53).	Algebra	Systems & Algebraic Manipulation	1
2022 AIME I	Find three-digit integer equal to its base-9 reversal.	Number Theory	Base Representations	2
2022 AIME I	Find distance between intersection points of angle bisectors in isosceles trapezoid.	Geometry	Triangles & Polygons	3
2022 AIME I	Count ordered pairs (r,s) up to 100 satisfying i*w^r = z^s.	Algebra	Complex Numbers	4
2022 AIME I	Find distance D between swimmers who meet at north bank simultaneously.	Algebra	Systems & Algebraic Manipulation	5
2022 AIME I	Count integer pairs (a,b) making sequence strictly increasing with no 4-term AP.	Combinatorics	Counting	6
2022 AIME I	Minimize positive fraction with distinct digits 1-9 in numerator and denominator.	Number Theory	Factors, Multiples & Divisibility	7
2022 AIME I	Find side length of small equilateral triangle formed by three tangent circles.	Geometry	Circular Geometry	8
2022 AIME I	Probability that arrangement of 12 colored blocks has even spacing between same colors.	Combinatorics	Counting	9
2022 AIME I	Find AC^2 given three tangent spheres cut by plane into congruent circles.	Geometry	3D Geometry & Solids	10
2022 AIME I	Find area of parallelogram given circle tangent to three sides and diagonal lengths.	Geometry	Triangles & Polygons	11
2022 AIME I	Compute ratio S_{2022}/S_{2021} for sum of \|A∩B\| over equal-size subsets.	Combinatorics	Counting	12
2022 AIME I	Count distinct numerators of repeating decimals 0.abcd in lowest terms.	Number Theory	Modular Arithmetic	13
2022 AIME I	Find perimeter of triangle given splitting lines through midpoints intersect at 30°.	Geometry	Triangles & Polygons	14
2022 AIME I	Find [(1-x)(1-y)(1-z)]^2 given three equations with square roots.	Algebra	Systems & Algebraic Manipulation	15
2022 AIME II	Find minimum adults after bus arrives given fractions change.	Algebra	Systems & Algebraic Manipulation	1
2022 AIME II	Probability Carl wins tennis tournament with given match odds.	Combinatorics	Probability & Expected Value	2
2022 AIME II	Find radius of sphere circumscribing a square pyramid.	Geometry	3D Geometry & Solids	3
2022 AIME II	Solve logarithmic equation for value expressed as log base 10.	Algebra	Logarithms & Exponents	4
2022 AIME II	Count triangles formed by connecting points with prime differences.	Combinatorics	Counting	5
2022 AIME II	Maximize difference between two ordered real numbers with constraints.	Algebra	Inequalities & Optimization	6
2022 AIME II	Find area bounded by three common tangents of two circles.	Geometry	Circular Geometry	7
2022 AIME II	Count numbers uniquely determined by floor values of n/4, n/5, n/6.	Number Theory	Modular Arithmetic	8
2022 AIME II	Count bounded regions formed by segments connecting points on parallel lines.	Combinatorics	Counting	9
2022 AIME II	Find remainder of sum of nested binomial coefficients modulo 1000.	Combinatorics		10
2022 AIME II	Find square of area of quadrilateral with given sides and angle bisectors.	Geometry	Triangles & Polygons	11
2022 AIME II	Minimize sum a+b given two ellipse equations with parameters.	Algebra	Inequalities & Optimization	12
2022 AIME II	Find coefficient of x^2022 in polynomial defined by rational expression.	Combinatorics	Counting	13
2022 AIME II	Find sum of three least c values with minimum stamp count 97.	Number Theory	Diophantine Equations	14
2022 AIME II	Find area of hexagon formed by two tangent circles and a third circle.	Geometry	Circular Geometry	15
2021 AIME I	Probability Zou wins exactly 5 of 6 races with win/loss dependence	Combinatorics	Probability & Expected Value	1
2021 AIME I	Area of intersection of two rectangles with given side lengths	Geometry	Triangles & Polygons	2
2021 AIME I	Number of positive integers less than 1000 as difference of powers of 2	Number Theory	Base Representations	3
2021 AIME I	Number of ways 66 identical coins split into 3 strictly increasing piles	Combinatorics	Counting	4
2021 AIME I	Sum of third terms of special arithmetic sequences of integers	Algebra	Polynomials	5
2021 AIME I	Find AP given distances from P to vertices of a cube	Geometry	3D Geometry & Solids	6
2021 AIME I	Number of pairs (m,n) with 1≤m<n≤30 such that sin(mx)+sin(nx)=2	Geometry	Trigonometry	7
2021 AIME I	Number of integers c such that equation has 12 distinct real solutions	Algebra	Functions	8
2021 AIME I	Area of isosceles trapezoid given distances from A to lines	Geometry	Triangles & Polygons	9
2021 AIME I	Sum of indices j where a_j = t/(t+1) in defined sequence	Number Theory	Modular Arithmetic	10
2021 AIME I	Perimeter of quadrilateral formed by feet of perpendiculars in cyclic quadrilateral	Geometry	Triangles & Polygons	11
2021 AIME I	Expected minutes until 2 of 3 frogs on dodecagon meet	Combinatorics	Probability & Expected Value	12
2021 AIME I	Distance between centers of two circles given third tangent circle conditions	Geometry	Circular Geometry	13
2021 AIME I	Least n such that σ(a^n)-1 divisible by 2021 for all positive a	Number Theory	Modular Arithmetic	14
2021 AIME I	Sum of least and greatest k with 4 intersection points on circle radius ≤21	Algebra	Systems & Algebraic Manipulation	15
2021 AIME II	Find the arithmetic mean of all three-digit palindromes.	Number Theory	Base Representations	1
2021 AIME II	Find AF in an equilateral triangle with given area ratio.	Geometry	Triangles & Polygons	2
2021 AIME II	Count permutations of 1-5 making a sum divisible by 3.	Combinatorics	Counting	3
2021 AIME II	Find m+n given shared complex root of two cubics.	Algebra	Complex Numbers	4
2021 AIME II	Find a^2+b^2 for s where obtuse triangles are unique.	Geometry	Trigonometry	5
2021 AIME II	Count ordered pairs of subsets satisfying a cardinality equation.	Combinatorics	Counting	6
2021 AIME II	Find m+n from a system of symmetric equations in a,b,c,d.	Algebra	Systems & Algebraic Manipulation	7
2021 AIME II	Probability ant is on top face after 8 moves on cube.	Combinatorics	Recursion in Combinatorics	8
2021 AIME II	Count ordered pairs (m,n) with gcd(2^m+1,2^n-1) > 1.	Number Theory	Modular Arithmetic	9
2021 AIME II	Find distance from line to tangency point of three spheres.	Geometry	3D Geometry & Solids	10
2021 AIME II	Find sum of greatest elements of possible sets S.	Number Theory	Modular Arithmetic	11
2021 AIME II	Find tan of acute angle between diagonals of quadrilateral.	Geometry	Trigonometry	12
2021 AIME II	Find least n such that 2^n+5^n-n is divisible by 1000.	Number Theory	Modular Arithmetic	13
2021 AIME II	Find angle BAC given ratios of angles in triangle geometry.	Geometry	Triangles & Polygons	14
2021 AIME II	Find least n such that f(n)/g(n) = 4/7 for recursive functions.	Algebra	Functions	15
2020 AIME I	Isosceles triangle with equal segments, find angle measure sum.	Geometry	Triangles & Polygons	1
2020 AIME I	Logarithms in geometric progression, find rational x.	Algebra	Logarithms & Exponents	2
2020 AIME I	Number with base-11 and base-8 representations, find least N.	Number Theory	Base Representations	3
2020 AIME I	Numbers ending in 2020 where prefix divides the number.	Number Theory	Factors, Multiples & Divisibility	4
2020 AIME I	Arrangements of 1-6 where removing one yields ascending/descending order.	Combinatorics	Counting	5
2020 AIME I	Two spheres in holes of radii 1 and 2, tangent to each other.	Geometry	3D Geometry & Solids	6
2020 AIME I	Committee with one more woman than men, count prime factors of N.	Combinatorics	Counting	7
2020 AIME I	Bug walks half as far each day, rotates 60°, find OP^2.	Algebra	Sequences & Series	8
2020 AIME I	Three divisors of 20^9 chosen with replacement, divisibility probability.	Combinatorics	Probability & Expected Value	9
2020 AIME I	Positive integers m,n with gcd(m+n,210)=1 and m^m multiple of n^n.	Number Theory	Modular Arithmetic	10
2020 AIME I	Quadratic f,g with integer coefficients, g(f(2))=g(f(4))=0.	Algebra	Polynomials	11
2020 AIME I	Least n such that 149^n-2^n divisible by 3^35^57^7.	Number Theory	Factors, Multiples & Divisibility	12
2020 AIME I	Triangle with angle bisector AD, perpendicular bisector meets other bisectors.	Geometry	Triangles & Polygons	13
2020 AIME I	Quadratic P with P(P(x))=0 having roots 3,4,a,b, find sum of (a+b)^2.	Algebra	Polynomials	14
2020 AIME I	Acute triangle ABC, tangent to circumcircle of HBC meets circumcircle at X,Y.	Geometry	Circular Geometry	15
2020 AIME II	Find ordered pairs (m,n) of positive integers with m^2 n = 20^20.	Number Theory	Factors, Multiples & Divisibility	1
2020 AIME II	Probability slope from random point to (5/8,3/8) is at least 1/2.	Geometry	Coordinate Geometry	2
2020 AIME II	Solve log_{2^x} 3^20 = log_{2^{x+3}} 3^2020 for x.	Algebra	Logarithms & Exponents	3
2020 AIME II	Find m+x+y for rotation mapping triangle ABC to A'B'C'.	Geometry	Triangles & Polygons	4
2020 AIME II	Least n where base-16 digit sum of g(n) uses digit A.	Number Theory	Base Representations	5
2020 AIME II	Find t_{2020} given recurrence t_n = (5t_{n-1}+1)/(25t_{n-2}).	Algebra	Sequences & Series	6
2020 AIME II	Max r^2 for sphere inside two intersecting right circular cones.	Geometry	3D Geometry & Solids	7
2020 AIME II	Least n so sum of zeros of f_n exceeds 500,000.	Algebra	Functions	8
2020 AIME II	Seating orders with no original adjacent pairs sitting together.	Combinatorics	Counting	9
2020 AIME II	Sum of n where sum of cubes divided by n+5 leaves remainder 17.	Number Theory	Modular Arithmetic	10
2020 AIME II	Find R(0) given quadratic polynomials with common root conditions.	Algebra	Polynomials	11
2020 AIME II	Count (m,n) so line through squares 200 and 2000 hits square 1099.	Combinatorics	Counting	12
2020 AIME II	Area of convex pentagon with incircle and given side lengths.	Geometry	Triangles & Polygons	13
2020 AIME II	Number of solutions to f(f(f(x)))=17 for 0≤x≤2020.	Algebra	Functions	14
2020 AIME II	Find XY^2 given tangents, projections, and sum of squares.	Geometry	Triangles & Polygons	15
2019 AIME I	Sum of 321 numbers 9, 99, 999, ...; find sum of digits of N.	Algebra	Sequences & Series	1
2019 AIME I	Probability that B-J >= 2 for distinct numbers 1-20.	Combinatorics	Probability & Expected Value	2
2019 AIME I	Area of hexagon formed by points on sides of right triangle.	Geometry	Triangles & Polygons	3
2019 AIME I	Number of ways to make up to 3 substitutions in soccer.	Combinatorics	Counting	4
2019 AIME I	Probability particle starting at (4,4) first hits axes at origin.	Combinatorics	Probability & Expected Value	5
2019 AIME I	Find MO in quadrilateral with perpendicular sides and diagonals.	Geometry	Triangles & Polygons	6
2019 AIME I	Solve system of log equations for x,y; find 3m+2n.	Algebra	Logarithms & Exponents	7
2019 AIME I	Given sin^10 x + cos^10 x = 11/36, find sin^12 x + cos^12 x.	Algebra	Polynomials	8
2019 AIME I	Sum of six least n with tau(n)+tau(n+1)=7.	Number Theory	Factors, Multiples & Divisibility	9
2019 AIME I	Absolute value of sum of pairwise products of distinct roots.	Algebra	Polynomials	10
2019 AIME I	Minimum perimeter of isosceles triangle with given excircle tangency.	Geometry	Triangles & Polygons	11
2019 AIME I	Find m+n for z = m+sqrt(n)+11i forming right triangle with f(z).	Algebra	Complex Numbers	12
2019 AIME I	Find BE in triangle with side lengths 4,5,6 and intersecting circles.	Geometry	Circular Geometry	13
2019 AIME I	Least odd prime factor of 2019^8 + 1.	Number Theory	Modular Arithmetic	14
2019 AIME I	Find PQ^2 for intersecting circles with chord AB and point P.	Geometry	Circular Geometry	15
2019 AIME II	Two congruent triangles overlap; find area of intersection.	Geometry	Triangles & Polygons	1
2019 AIME II	Frog jumps on lily pads; probability it visits pad 7.	Combinatorics	Probability & Expected Value	2
2019 AIME II	Count 7-tuples of positive integers satisfying three product equations.	Number Theory	Factors, Multiples & Divisibility	3
2019 AIME II	Probability product of four dice rolls is a perfect square.	Combinatorics	Probability & Expected Value	4
2019 AIME II	Seat ambassadors and advisors at a round table with constraints.	Combinatorics	Counting	5
2019 AIME II	Solve system of log equations for base b.	Algebra	Logarithms & Exponents	6
2019 AIME II	Find perimeter of triangle formed by lines parallel to sides.	Geometry	Triangles & Polygons	7
2019 AIME II	Evaluate polynomial at complex root; find f(1) mod 1000.	Algebra	Complex Numbers	8
2019 AIME II	Sum of 20-pretty positive integers less than 2019.	Number Theory	Factors, Multiples & Divisibility	9
2019 AIME II	Find angle theta with alternating sign pattern of tan(2^n theta).	Geometry	Trigonometry	10
2019 AIME II	Find AK where circles tangent to sides intersect again.	Geometry	Circular Geometry	11
2019 AIME II	Count progressive sequences with sum 360.	Combinatorics	Recursion in Combinatorics	12
2019 AIME II	Find n such that area bounded by PA6, PA7, arc is 1/8 - sqrt2/n.	Geometry	Coordinate Geometry	13
2019 AIME II	Find sum of n where 91 is largest unformable postage.	Number Theory	Modular Arithmetic	14
2019 AIME II	Find AB*AC given lengths on orthic triangle chord.	Geometry	Triangles & Polygons	15
2018 AIME I	Count ordered pairs (a,b) so x^2+ax+b factors over integers.	Combinatorics	Counting	1
2018 AIME I	Find base-10 number n given its base-14, base-15, base-6 representations.	Number Theory	Base Representations	2
2018 AIME I	Probability that first 5 cards have all reds and greens adjacent.	Combinatorics	Probability & Expected Value	3
2018 AIME I	Find AD in isosceles triangle ABC with AD = DE = EC.	Geometry	Triangles & Polygons	4
2018 AIME I	Find product of all possible K satisfying logarithmic equations.	Algebra	Logarithms & Exponents	5
2018 AIME I	Count complex z with \|z\|=1 and z^720 - z^120 real.	Algebra	Complex Numbers	6
2018 AIME I	Count isosceles triangles from vertices of a right hexagonal prism.	Combinatorics	Counting	7
2018 AIME I	Find d^2 for largest circle inside equiangular hexagon with given sides.	Geometry	Triangles & Polygons	8
2018 AIME I	Count 4-element subsets of {1..20} with sums 16 and 24.	Combinatorics	Counting	9
2018 AIME I	Count 15-step bug paths on wheel starting and ending at A.	Combinatorics	Recursion in Combinatorics	10
2018 AIME I	Least n such that 3^n ends in 01 in base 143.	Number Theory	Modular Arithmetic	11
2018 AIME I	Probability sum of random subset of {1..18} is divisible by 3.	Combinatorics	Probability & Expected Value	12
2018 AIME I	Minimize area of triangle formed by incenters of two subtriangles.	Geometry	Triangles & Polygons	13
2018 AIME I	Count sequences of ≤12 jumps on heptagon ending at E.	Combinatorics	Recursion in Combinatorics	14
2018 AIME I	Find common area K of three cyclic quadrilaterals with given sine values.	Geometry	Triangles & Polygons	15
2018 AIME II	Three runners start at points A, B, C on a line; find AB.	Algebra	Systems & Algebraic Manipulation	1
2018 AIME II	Find product of three terms of a recursive sequence modulo 11.	Algebra	Sequences & Series	2
2018 AIME II	Find sum of bases b where 36_b is square and 27_b is cube.	Number Theory	Base Representations	3
2018 AIME II	Find area enclosed by self-intersecting octagon CORNELIA.	Geometry	Coordinate Geometry	4
2018 AIME II	Find a^2+b^2 given complex numbers x,y,z with given products.	Algebra	Complex Numbers	5
2018 AIME II	Probability that polynomial roots are all real for random a.	Algebra	Polynomials	6
2018 AIME II	Count segments with rational length dividing triangle into equal-area regions.	Geometry	Triangles & Polygons	7
2018 AIME II	Number of jump sequences from (0,0) to (4,4) with step sizes 1 or 2.	Combinatorics	Counting	8
2018 AIME II	Find area of polygon formed by centroids of 7 triangles in octagon.	Geometry	Coordinate Geometry	9
2018 AIME II	Number of functions f: {1..5}->{1..5} with f(f(x))=f(f(f(x))).	Combinatorics	Counting	10
2018 AIME II	Number of permutations of 1..6 with a condition on first k terms.	Combinatorics	Counting	11
2018 AIME II	Find area of convex quadrilateral given side lengths and area condition.	Geometry	Triangles & Polygons	12
2018 AIME II	Probability of rolling an odd number of times to get 1-2-3.	Combinatorics	Probability & Expected Value	13
2018 AIME II	Find AQ given incircle tangency conditions and side lengths.	Geometry	Triangles & Polygons	14
2018 AIME II	Number of functions f: {0..6}->Z with f(0)=0, f(6)=12, and Lipschitz condition.	Combinatorics	Counting	15
2017 AIME I	Count triangles with positive area from 15 points on triangle sides.	Combinatorics	Counting	1
2017 AIME I	Find m+n+r+s given constant remainders for two sets of numbers.	Number Theory	Modular Arithmetic	2
2017 AIME I	Find remainder of sum of units digits of triangular numbers.	Number Theory	Modular Arithmetic	3
2017 AIME I	Find volume of pyramid with given base and equal lateral edges.	Geometry	3D Geometry & Solids	4
2017 AIME I	Find base-ten number abc from base-8 and base-12 representations.	Number Theory	Base Representations	5
2017 AIME I	Find difference between possible vertex angles given chord probability.	Combinatorics	Probability & Expected Value	6
2017 AIME I	Sum of triple binomial product over nonnegative a,b with a+b≤6.	Combinatorics	Counting	7
2017 AIME I	Probability that chord QR ≤100 given random angles a,b.	Combinatorics	Probability & Expected Value	8
2017 AIME I	Find least n>10 such that recursive sequence a_n is multiple of 99.	Number Theory	Modular Arithmetic	9
2017 AIME I	Find real part of z with maximal imaginary part given complex condition.	Algebra	Complex Numbers	10
2017 AIME I	Count 3x3 arrangements where median of row medians equals 5.	Combinatorics	Counting	11
2017 AIME I	Count product-free subsets of {1,...,10}.	Combinatorics	Counting	12
2017 AIME I	Sum of Q(m) for m=2..2017, where Q(m) is least n with cube property.	Number Theory	Modular Arithmetic	13
2017 AIME I	Find x mod 1000 given logarithmic equations with base a.	Algebra	Logarithms & Exponents	14
2017 AIME I	Find minimal area of equilateral triangle inscribed in right triangle.	Geometry	Triangles & Polygons	15
2017 AIME II	Count subsets of {1..8} not subsets of two given 5-element sets.	Combinatorics	Counting	1
2017 AIME II	Probability that team T4 wins a 4-team playoff with given win probabilities.	Combinatorics	Probability & Expected Value	2
2017 AIME II	Probability a random point in triangle is closer to vertex B than A or C.	Geometry	Coordinate Geometry	3
2017 AIME II	Count positive integers ≤2017 with no digit 0 in base 3.	Number Theory	Base Representations	4
2017 AIME II	Maximize sum of two unknown pairwise sums given four known sums.	Algebra	Systems & Algebraic Manipulation	5
2017 AIME II	Sum of positive integers n making sqrt(n^2+85n+2017) an integer.	Number Theory	Diophantine Equations	6
2017 AIME II	Count integer k in [-500,500] for which log(kx)=2log(x+2) has one real solution.	Algebra	Logarithms & Exponents	7
2017 AIME II	Count n<2017 making 1+n+n^2/2!+...+n^6/6! an integer.	Number Theory	Modular Arithmetic	8
2017 AIME II	Probability Sharon can discard a card and still have all colors and numbers.	Combinatorics	Probability & Expected Value	9
2017 AIME II	Area of triangle CDP given rectangle and bisecting segment BP.	Geometry	Coordinate Geometry	10
2017 AIME II	Number of ways to orient roads so every town reachable from every other.	Combinatorics	Counting	11
2017 AIME II	Distance from center of C0 to point B inside all nested circles.	Geometry	Triangles & Polygons	12
2017 AIME II	Sum of n where f(n+1)=f(n)+78 for isosceles triangle count in n-gon.	Combinatorics	Counting	13
2017 AIME II	Number of lines containing exactly 8 points in a 10x10x10 grid.	Combinatorics	Counting	14
2017 AIME II	Minimum possible sum of distances from a point to tetrahedron vertices.	Geometry	3D Geometry & Solids	15
2016 AIME I	Sum of infinite geometric series S(a) and S(-a) given product 2016.	Algebra	Sequences & Series	1
2016 AIME I	Probability of rolling a 7 with weighted dice.	Combinatorics	Probability & Expected Value	2
2016 AIME I	Number of paths from top to bottom of icosahedron without repeating vertices.	Combinatorics	Counting	3
2016 AIME I	Find h^2 given dihedral angle 60° in a right prism pyramid.	Geometry	3D Geometry & Solids	4
2016 AIME I	Find n+t given arithmetic sequences of pages and minutes summing to 374 and 319.	Algebra	Sequences & Series	5
2016 AIME I	Find IC in triangle with incenter I and angle bisector meeting circumcircle at D.	Geometry	Triangles & Polygons	6
2016 AIME I	Number of integer pairs (a,b) making complex number real.	Algebra	Complex Numbers	7
2016 AIME I	Minimum sum of three 3-digit numbers from permutation of digits 1-9 with units digit 0.	Combinatorics	Counting	8
2016 AIME I	Maximum area of rectangle AQRS containing triangle ABC with given sides and sin A.	Geometry	Trigonometry	9
2016 AIME I	Find a1 given a13=2016 in a sequence alternating geometric and arithmetic.	Algebra	Sequences & Series	10
2016 AIME I	Find P(7/2) given functional equation (x-1)P(x+1)=(x+2)P(x) and P(2)^2=P(3).	Algebra	Polynomials	11
2016 AIME I	Least positive integer m such that m^2-m+11 is product of at least 4 primes.	Number Theory	Factors, Multiples & Divisibility	12
2016 AIME I	Expected number of jumps for frog to reach river y=24 from (0,21) with fence at y=0.	Combinatorics	Probability & Expected Value	13
2016 AIME I	Number of squares and circles intersected by line from (0,0) to (1001,429).	Geometry	Coordinate Geometry	14
2016 AIME I	Find AB^2 given intersecting circles, tangents, and collinear points C, Y, D.	Geometry	Circular Geometry	15
2016 AIME II	Three people have peanuts in geometric progression; after eating, they form an arithmetic progression.	Algebra	Systems & Algebraic Manipulation	1
2016 AIME II	Find probability it rains at least one weekend day given conditional probabilities.	Combinatorics	Probability & Expected Value	2
2016 AIME II	Solve a system of logarithmic equations for absolute values of logs.	Algebra	Logarithms & Exponents	3
2016 AIME II	Find minimum volume of a rectangular box with colored unit cubes.	Number Theory	Factors, Multiples & Divisibility	4
2016 AIME II	Find perimeter of right triangle with infinite altitude sum equal to 6 times perimeter.	Geometry	Triangles & Polygons	5
2016 AIME II	Find sum of absolute coefficients of product of polynomials.	Algebra	Polynomials	6
2016 AIME II	Find difference between max and min area of square inscribed in nested squares.	Geometry	Triangles & Polygons	7
2016 AIME II	Count sets of three distinct positive integers with given product.	Combinatorics	Counting	8
2016 AIME II	Find middle term of sum of arithmetic and geometric sequences given two terms.	Algebra	Sequences & Series	9
2016 AIME II	Find length ST in cyclic quadrilateral with given segment lengths.	Geometry	Circular Geometry	10
2016 AIME II	Count numbers less than 1000 that are neither 7-nice nor 8-nice.	Number Theory	Factors, Multiples & Divisibility	11
2016 AIME II	Count colorings of six-section ring with no adjacent same color.	Combinatorics	Counting	12
2016 AIME II	Find average score of rook placements on 6x6 board.	Combinatorics	Counting	13
2016 AIME II	Find distance from point O to vertices of equilateral triangle and two external points.	Geometry	3D Geometry & Solids	14
2016 AIME II	Maximize x_2 given sum and sum of pairwise products constraints.	Algebra	Inequalities & Optimization	15
2015 AIME I	Find the positive difference between two alternating sum expressions.	Algebra	Systems & Algebraic Manipulation	1
2015 AIME I	Probability exactly two of three random sleepers are from the same country.	Combinatorics	Probability & Expected Value	2
2015 AIME I	Find prime p such that 16p+1 is a perfect cube.	Number Theory	Diophantine Equations	3
2015 AIME I	Area of triangle formed by midpoints in an equilateral triangle configuration.	Geometry	Coordinate Geometry	4
2015 AIME I	Probability Wednesday is first day Sandy picks matching socks.	Combinatorics	Probability & Expected Value	5
2015 AIME I	Find angle measure given arc relationships in two circles.	Geometry	Circular Geometry	6
2015 AIME I	Area of a square inscribed in a square with midpoints.	Geometry	Triangles & Polygons	7
2015 AIME I	Smallest n with digit sum 20 and n+864 also digit sum 20.	Number Theory	Base Representations	8
2015 AIME I	Count triples (a1,a2,a3) where the generated sequence hits zero.	Combinatorics	Counting	9
2015 AIME I	Find \|f(0)\| for cubic f with six equal absolute values.	Algebra	Polynomials	10
2015 AIME I	Smallest perimeter of isosceles triangle with given angle bisector length.	Geometry	Triangles & Polygons	11
2015 AIME I	Arithmetic mean of least elements of all 1000-element subsets of {1..2015}.	Combinatorics	Counting	12
2015 AIME I	Product of cosecant squares of odd angles from 1 to 89 degrees.	Algebra	Complex Numbers	13
2015 AIME I	Count n where area under floor(sqrt(x)) from 1 to n is integer.	Algebra	Functions	14
2015 AIME I	Area of unpainted face after slicing a painted cylinder.	Geometry	3D Geometry & Solids	15
2015 AIME II	Find remainder when least N satisfying percent conditions is divided by 1000.	Number Theory	Modular Arithmetic	1
2015 AIME II	Probability a randomly chosen Latin student is a sophomore.	Combinatorics	Probability & Expected Value	2
2015 AIME II	Find least positive integer divisible by 17 with digit sum 17.	Number Theory	Modular Arithmetic	3
2015 AIME II	Find p+q for perimeter of isosceles trapezoid expressed as log.	Geometry	Triangles & Polygons	4
2015 AIME II	Least n so probability adjacent squares in n×n grid is less than 1/2015.	Combinatorics	Probability & Expected Value	5
2015 AIME II	Sum of two possible c values given polynomial with positive integer roots.	Algebra	Polynomials	6
2015 AIME II	Find m+n for coefficient beta in area expression of rectangle in triangle.	Geometry	Triangles & Polygons	7
2015 AIME II	Maximum value of rational expression given inequality on a and b.	Algebra	Inequalities & Optimization	8
2015 AIME II	Volume squared of water displaced by cube with vertical diagonal in cylinder.	Geometry	3D Geometry & Solids	9
2015 AIME II	Number of quasi-increasing permutations of integers 1 through 7.	Combinatorics	Recursion in Combinatorics	10
2015 AIME II	Find m+n for BP given circumcenter and perpendicular line in acute triangle.	Geometry	Triangles & Polygons	11
2015 AIME II	Number of 10-letter strings with no more than 3 adjacent identical letters.	Combinatorics	Recursion in Combinatorics	12
2015 AIME II	Index of 100th term where sum of sines from 1 to n is negative.	Algebra	Sequences & Series	13
2015 AIME II	Evaluate 2x^3 + (xy)^3 + 2y^3 given two symmetric equations.	Algebra	Systems & Algebraic Manipulation	14
2015 AIME II	Common area of two triangles formed by line through tangency point of circles.	Geometry	Circular Geometry	15
2014 AIME I	Minimum lace length for sneaker with 8 eyelets on a rectangle.	Geometry	Coordinate Geometry	1
2014 AIME I	Find N so probability both balls same color is 0.58.	Combinatorics	Probability & Expected Value	2
2014 AIME I	Number of reduced fractions with numerator and denominator summing to 1000.	Number Theory	Factors, Multiples & Divisibility	3
2014 AIME I	Length of trains given passing times for two cyclists.	Algebra	Systems & Algebraic Manipulation	4
2014 AIME I	Number of communal subsets of a regular 12-gon's vertices.	Combinatorics	Counting	5
2014 AIME I	Find h such that two parabolas have positive integer x-intercepts.	Algebra	Polynomials	6
2014 AIME I	Maximum tan^2 of argument of (w-z)/z with \|w\|=1, \|z\|=10.	Algebra	Complex Numbers	7
2014 AIME I	Four-digit number N such that N and N^2 end in same four digits.	Number Theory	Modular Arithmetic	8
2014 AIME I	Find x2(x1+x3) for three real roots of a cubic equation.	Algebra	Polynomials	9
2014 AIME I	Sin^2 of angle BEA after rolling a small disk around a larger one.	Geometry	Circular Geometry	10
2014 AIME I	Probability token ends on \|y\|=\|x\| after six random unit moves.	Combinatorics	Probability & Expected Value	11
2014 AIME I	Probability ranges of two random functions from {1,2,3,4} to itself are disjoint.	Combinatorics	Counting	12
2014 AIME I	Area of square given perpendicular segments EG=FH=34 and area ratios.	Geometry	Triangles & Polygons	13
2014 AIME I	Largest real solution to equation with four rational terms equals quadratic.	Algebra	Systems & Algebraic Manipulation	14
2014 AIME I	Length DE in triangle ABC with circle through B, D, E, F, G.	Geometry	Triangles & Polygons	15
2014 AIME II	Three painters with different rates paint a room; find total minutes.	Algebra	Systems & Algebraic Manipulation	1
2014 AIME II	Given probabilities of risk factors, find conditional probability as fraction sum.	Combinatorics	Probability & Expected Value	2
2014 AIME II	Rectangle sides pressed to form hexagon of equal area; find a^2.	Geometry	Triangles & Polygons	3
2014 AIME II	Sum of two repeating decimals equals 33/37; find three-digit abc.	Number Theory	Base Representations	4
2014 AIME II	Cubic polynomials share roots with shifts; find sum of absolute b values.	Algebra	Polynomials	5
2014 AIME II	Given two sixes from a fair or biased die, find probability of third six.	Combinatorics	Probability & Expected Value	6
2014 AIME II	Sum of logs of function with cosine exponent equals ±1; find sum of n.	Algebra	Logarithms & Exponents	7
2014 AIME II	Three tangent circles with given radii; find radius of circle D as sqrt(m)-n.	Geometry	Circular Geometry	8
2014 AIME II	Ten chairs in a circle; count subsets with at least three adjacent chairs.	Combinatorics	Counting	9
2014 AIME II	Complex numbers satisfying reciprocal equation; find area of polygon modulo 1000.	Algebra	Complex Numbers	10
2014 AIME II	Triangle with given angles and midpoint; find AE as (a-sqrt(b))/c.	Geometry	Triangles & Polygons	11
2014 AIME II	Triangle angles satisfy cos sum condition; find max possible third side length.	Geometry	Trigonometry	12
2014 AIME II	Random pairing of 10 left and right shoes; find probability of no small bad collection.	Combinatorics	Counting	13
2014 AIME II	Triangle with altitude, angle bisector, median; find AP^2 as reduced fraction.	Geometry	Triangles & Polygons	14
2014 AIME II	Sequence defined by smallest prime not dividing term; find t where x_t=2090.	Number Theory	Modular Arithmetic	15
2013 AIME I	Tom swims, bikes, and runs a triathlon; find biking time.	Algebra	Systems & Algebraic Manipulation	1
2013 AIME I	Count five-digit numbers divisible by 5 with equal first/last digits and digit sum divisible by 5.	Combinatorics	Counting	2
2013 AIME I	Square divided by lines into two squares and two rectangles; find sum of ratios.	Algebra	Systems & Algebraic Manipulation	3
2013 AIME I	Probability a 13-square coloring looks same after 90-degree rotation.	Combinatorics	Counting	4
2013 AIME I	Real root of cubic expressed as sum of cube roots; find sum of constants.	Algebra	Polynomials	5
2013 AIME I	Probability all 3 math textbooks end up in same box.	Combinatorics	Probability & Expected Value	6
2013 AIME I	Rectangular box with given width/length; triangle area of face centers is 30; find height.	Geometry	3D Geometry & Solids	7
2013 AIME I	Domain of arcsin(log) is interval of length 1/2013; find minimal m+n mod 1000.	Algebra	Functions	8
2013 AIME I	Equilateral triangle folded so vertex touches side; find fold line length.	Geometry	Triangles & Polygons	9
2013 AIME I	Complex root of cubic with integer coefficients; sum of all possible sums of zeros.	Algebra	Polynomials	10
2013 AIME I	Find sum of distinct prime divisors of least N satisfying block distribution conditions.	Number Theory	Modular Arithmetic	11
2013 AIME I	Triangle with hexagon inside; find area of triangle expressed as (a+b√c)/d.	Geometry	Triangles & Polygons	12
2013 AIME I	Triangle with nested similar triangles; find denominator of area of union.	Geometry	Triangles & Polygons	13
2013 AIME I	Infinite trigonometric series P and Q with given ratio; find sin θ.	Algebra	Sequences & Series	14
2013 AIME I	Count ordered triples (A,B,C) satisfying arithmetic sequence and divisibility conditions.	Number Theory	Modular Arithmetic	15
2013 AIME II	Convert 6:36 AM to new metric time with 10 hours and 100 minutes per day.	Algebra	Systems & Algebraic Manipulation	1
2013 AIME II	Find sum of all possible a+b given a logarithmic equation.	Algebra	Logarithms & Exponents	2
2013 AIME II	Find candle height after half the total burn time given burn rates.	Algebra	Sequences & Series	3
2013 AIME II	Find product of coordinates of center of equilateral triangle given two vertices.	Geometry	Coordinate Geometry	4
2013 AIME II	Find sine of angle DAE in equilateral triangle with trisected side.	Geometry	Trigonometry	5
2013 AIME II	Find least N such that 1000N to 1000N+999 contains no perfect square.	Number Theory	Modular Arithmetic	6
2013 AIME II	Find number of files sorted in first 1.5 hours given reassignment pattern.	Algebra	Systems & Algebraic Manipulation	7
2013 AIME II	Find radius of circle circumscribing hexagon with given side lengths.	Geometry	Circular Geometry	8
2013 AIME II	Count tilings of 7x1 board with colored tiles using all three colors at least once.	Combinatorics	Counting	9
2013 AIME II	Find maximum area of triangle BKL given circle and external point A.	Geometry	Circular Geometry	10
2013 AIME II	Count functions f from 7-element set to itself such that f(f(x)) is constant.	Combinatorics	Counting	11
2013 AIME II	Count cubic polynomials with integer coefficients whose roots have magnitude 13 or 20.	Algebra	Polynomials	12
2013 AIME II	Find area of isosceles triangle given lengths from midpoint and trisection point.	Geometry	Triangles & Polygons	13
2013 AIME II	Sum of maximum remainders when n is divided by k for k up to n/2, n=20 to 100.	Number Theory	Modular Arithmetic	14
2013 AIME II	Find value of trigonometric expression given two equations involving triangle angles.	Geometry	Trigonometry	15
2012 AIME I	Count three-digit numbers abc where abc and cba are multiples of 4.	Number Theory	Modular Arithmetic	1
2012 AIME I	Find sum of first, last, and middle terms of an arithmetic sequence.	Algebra	Sequences & Series	2
2012 AIME I	Count ways waiter serves meals so exactly one person gets correct meal.	Combinatorics	Counting	3
2012 AIME I	Find n+t when Butch and Sundance first meet at a milepost.	Algebra	Systems & Algebraic Manipulation	4
2012 AIME I	Count subtractions of binary numbers with 5 zeros and 8 ones giving 1.	Combinatorics	Counting	5
2012 AIME I	Find n where imaginary part of z is sin(mπ/n) given z^13=w and w^11=z.	Algebra	Complex Numbers	6
2012 AIME I	Find coins the center student had originally in a network of 16 students.	Algebra	Systems & Algebraic Manipulation	7
2012 AIME I	Find p+q where volume of larger solid from plane cut is p/q.	Geometry	3D Geometry & Solids	8
2012 AIME I	Find p+q where xy^5z = 1/2^(p/q) given three equal log expressions.	Algebra	Logarithms & Exponents	9
2012 AIME I	Find remainder when 10th smallest element of T is divided by 1000.	Number Theory	Modular Arithmetic	10
2012 AIME I	Count reachable points (x,y) with \|x\|+\|y\|≤100 via given jumps.	Combinatorics	Counting	11
2012 AIME I	Find tan B in right triangle with CD and CE trisecting angle C.	Geometry	Triangles & Polygons	12
2012 AIME I	Find largest area of equilateral triangle with vertices on three concentric circles.	Geometry	Triangles & Polygons	13
2012 AIME I	Find h^2 where h is hypotenuse of right triangle from roots of cubic.	Algebra	Complex Numbers	14
2012 AIME I	Count n<1000 where a exists satisfying seating rearrangement conditions.	Number Theory	Modular Arithmetic	15
2012 AIME II	Find number of positive integer pairs (m,n) solving 20m+12n=2012.	Number Theory	Modular Arithmetic	1
2012 AIME II	Find a_9 given two geometric sequences with same ratio and a_15=b_11.	Algebra	Sequences & Series	2
2012 AIME II	Count committees of 6 with 3 men, 3 women, 2 from each of 3 departments.	Combinatorics	Counting	3
2012 AIME II	Find p+q+r for ratio of field length, width, and distance to point D.	Algebra	Systems & Algebraic Manipulation	4
2012 AIME II	Find volume of pyramid formed by folding star figure cut from square.	Geometry	3D Geometry & Solids	5
2012 AIME II	Find c+d for z^4=c+di given \|z\|=5 and distance maximized.	Algebra	Complex Numbers	6
2012 AIME II	Find remainder when 1000th number with 8 binary ones is divided by 1000.	Combinatorics	Counting	7
2012 AIME II	Find smallest possible \|zw\|^2 given system with complex numbers z and w.	Algebra	Complex Numbers	8
2012 AIME II	Find p+q for sin2x/sin2y + cos2x/cos2y = p/q given trig ratios.	Geometry	Trigonometry	9
2012 AIME II	Count positive integers n<1000 with positive real x such that n=x floor(x).	Algebra	Functions	10
2012 AIME II	Find m+n for x=m/n satisfying f_{1001}(x)=x-3 with recursive function definition.	Algebra	Functions	11
2012 AIME II	Count numbers ≤10000 that are 7-safe, 11-safe, and 13-safe.	Number Theory	Modular Arithmetic	12
2012 AIME II	Find sum of squares of distances from C to four points E_k.	Geometry	Trigonometry	13
2012 AIME II	Count handshake arrangements where each of 9 people shakes exactly 2 hands.	Combinatorics	Counting	14
2012 AIME II	Find m+n for AF^2=m/n in triangle with inscribed circle and angle bisector.	Geometry	Triangles & Polygons	15
2011 AIME I	Mixing acid solutions from three jars to achieve 50% concentration.	Algebra	Systems & Algebraic Manipulation	1
2011 AIME I	Find length EF inside rectangle with parallel segments.	Geometry	Triangles & Polygons	2
2011 AIME I	Find sum of coordinates of point in rotated coordinate system.	Geometry	Coordinate Geometry	3
2011 AIME I	Find distance between feet of perpendiculars from vertex to angle bisectors.	Geometry	Triangles & Polygons	4
2011 AIME I	Count arrangements of digits 1-9 on nonagon with sum condition.	Combinatorics	Counting	5
2011 AIME I	Find minimum a for parabola with integer value at x=1.	Algebra	Polynomials	6
2011 AIME I	Count m such that a power equals sum of 2011 other powers.	Number Theory	Modular Arithmetic	7
2011 AIME I	Find maximum table height from folded triangle with parallel top.	Geometry	Triangles & Polygons	8
2011 AIME I	Solve logarithmic equation for cot^2 x in given interval.	Algebra	Logarithms & Exponents	9
2011 AIME I	Find sum of n for which obtuse triangle probability is 93/125.	Combinatorics	Probability & Expected Value	10
2011 AIME I	Sum of all distinct remainders of 2^n modulo 1000.	Number Theory	Modular Arithmetic	11
2011 AIME I	Find least women so probability of 4+ consecutive men ≤ 1%.	Combinatorics	Counting	12
2011 AIME I	Distance from cube vertex to plane given three adjacent heights.	Geometry	3D Geometry & Solids	13
2011 AIME I	Find cos of angle in octagon with perpendicular rays.	Geometry	Coordinate Geometry	14
2011 AIME I	Find sum of absolute values of integer roots of cubic.	Algebra	Polynomials	15
2011 AIME II	Fraction of beverage drunk given wasted fraction condition	Algebra	Systems & Algebraic Manipulation	1
2011 AIME II	Area of square given three equal segments from sides	Geometry	Triangles & Polygons	2
2011 AIME II	Smallest angle of convex 18-gon with arithmetic sequence angles	Algebra	Sequences & Series	3
2011 AIME II	Ratio CP to PA in triangle with angle bisector and midpoint	Geometry	Triangles & Polygons	4
2011 AIME II	Sum of first 6033 terms of geometric sequence given partial sums	Algebra	Sequences & Series	5
2011 AIME II	Number of ordered quadruples with a+d > b+c from 1 to 10	Combinatorics	Counting	6
2011 AIME II	Maximum red marbles and arrangements with equal neighbor color counts	Combinatorics	Counting	7
2011 AIME II	Maximum real part of sum of 12th roots of unity scaled by i	Algebra	Complex Numbers	8
2011 AIME II	Maximum sum of triple products given sum and product constraint	Algebra	Inequalities & Optimization	9
2011 AIME II	Distance from center to intersection of chords in a circle	Geometry	Circular Geometry	10
2011 AIME II	Sum of infinite series involving determinants of tridiagonal matrices	Algebra	Sequences & Series	11
2011 AIME II	Probability each delegate sits next to someone from another country	Combinatorics	Probability & Expected Value	12
2011 AIME II	Length AP on square diagonal given circumcenter angle condition	Geometry	Triangles & Polygons	13
2011 AIME II	Number of permutations of 1..30 with divisibility constraints on positions	Combinatorics	Counting	14
2011 AIME II	Probability floor of sqrt(P(x)) equals sqrt(P(floor(x))) for random x	Algebra	Functions	15
2010 AIME I	Probability that exactly one of two divisors of 2010^2 is a perfect square.	Combinatorics	Probability & Expected Value	1
2010 AIME I	Remainder when product of numbers 9, 99, 999, ... is divided by 1000.	Number Theory	Modular Arithmetic	2
2010 AIME I	Find x+y given y = 3x/4 and x^y = y^x.	Algebra	Logarithms & Exponents	3
2010 AIME I	Probability Jackie and Phil get same number of heads with biased coin.	Combinatorics	Probability & Expected Value	4
2010 AIME I	Number of possible values of a given a > b > c > d and sum conditions.	Algebra	Systems & Algebraic Manipulation	5
2010 AIME I	Find P(16) for quadratic P(x) bounded by two quadratics with P(11)=181.	Algebra	Polynomials	6
2010 AIME I	Number of ordered triples of subsets of {1..7} with pairwise intersection 1 and empty triple intersection.	Combinatorics	Counting	7
2010 AIME I	Minimum radius of disk containing region defined by floor(x)^2 + floor(y)^2 = 25.	Geometry	Coordinate Geometry	8
2010 AIME I	Greatest possible value of a^3+b^3+c^3 given system x^3-xyz=2,6,20.	Algebra	Systems & Algebraic Manipulation	9
2010 AIME I	Number of ways to write 2010 as a_310^3 + a_210^2 + a_1*10 + a_0 with 0 ≤ a_i ≤ 99.	Number Theory	Base Representations	10
2010 AIME I	Volume of solid formed by revolving region R around line 3y - x = 15.	Geometry	3D Geometry & Solids	11
2010 AIME I	Smallest m such that any partition of {3,4,...,m} has a subset with a,b,c where ab=c.	Combinatorics	Counting	12
2010 AIME I	Find DA given rectangle and semicircle divided by line with area ratio 1:2.	Geometry	Triangles & Polygons	13
2010 AIME I	Largest n such that sum_{k=1}^{100} floor(log10(kn)) ≤ 300.	Algebra	Functions	14
2010 AIME I	Ratio AM/CM in triangle ABC where incircles of ABM and BCM have equal radii.	Geometry	Triangles & Polygons	15
2010 AIME II	Greatest integer multiple of 36 with distinct even digits, remainder mod 1000.	Number Theory	Modular Arithmetic	1
2010 AIME II	Probability distance to closest side of unit square is between 1/5 and 1/3.	Combinatorics	Probability & Expected Value	2
2010 AIME II	Greatest n such that 2^n divides product of differences of integers 1 to 20.	Number Theory	Factors, Multiples & Divisibility	3
2010 AIME II	Probability Dave walks at most 400 feet after random gate change.	Combinatorics	Probability & Expected Value	4
2010 AIME II	Find sqrt of sum of squares of logs given product and sum of products.	Algebra	Logarithms & Exponents	5
2010 AIME II	Smallest n so x^4 - nx + 63 factors into nonconstant integer polynomials.	Algebra	Polynomials	6
2010 AIME II	Find \|a+b+c\| for cubic with given complex roots involving w.	Algebra	Complex Numbers	7
2010 AIME II	Number of ordered pairs of disjoint nonempty sets covering 1..12 with size constraints.	Combinatorics	Counting	8
2010 AIME II	Ratio of area of smaller hexagon to larger regular hexagon.	Geometry	Triangles & Polygons	9
2010 AIME II	Number of second-degree integer polynomials with integer zeros and f(0)=2010.	Algebra	Polynomials	10
2010 AIME II	Number of 3x3 matrices with five 1's and four 0's, no two full lines equal.	Combinatorics	Counting	11
2010 AIME II	Minimum common perimeter of two noncongruent isosceles triangles with same area.	Geometry	Triangles & Polygons	12
2010 AIME II	Minimum probability Alex and Dylan are on same team given they hold a and a+9.	Combinatorics	Probability & Expected Value	13
2010 AIME II	Ratio AP/BP in right triangle with given angle condition and CP=1.	Geometry	Triangles & Polygons	14
2010 AIME II	Ratio BQ/CQ where Q is intersection of ray AP with BC in triangle ABC.	Geometry	Triangles & Polygons	15
2009 AIME I	Find difference between largest and smallest 3-digit numbers with digits in geometric sequence.	Number Theory	Base Representations	1
2009 AIME I	Find n such that z/(z+n)=4i with Im(z)=164.	Algebra	Complex Numbers	2
2009 AIME I	Find p+q given p=m/n and probability of 3 heads equals 1/25 of 5 heads.	Combinatorics	Probability & Expected Value	3
2009 AIME I	Find AC/AP in parallelogram with points M on AB and N on AD.	Geometry	Triangles & Polygons	4
2009 AIME I	Find LP in triangle with given side lengths and angle bisector.	Geometry	Triangles & Polygons	5
2009 AIME I	Count positive integers N<1000 such that x^{floor(x)}=N has a solution.	Algebra	Functions	6
2009 AIME I	Find least k>1 such that a_k is integer given recurrence with exponent.	Algebra	Sequences & Series	7
2009 AIME I	Sum of all positive differences of pairs from {2^0,...,2^10} mod 1000.	Algebra	Sequences & Series	8
2009 AIME I	Number of guesses for three prizes given digits 1,1,1,1,3,3,3.	Combinatorics	Counting	9
2009 AIME I	Number of seating arrangements for 15 committee members with restrictions.	Combinatorics	Counting	10
2009 AIME I	Number of triangles OPQ with integer area, P,Q on line 41x+y=2009.	Geometry	Coordinate Geometry	11
2009 AIME I	Ratio of perimeter of triangle ABI to AB in right triangle with incircle.	Geometry	Triangles & Polygons	12
2009 AIME I	Minimum possible a1+a2 for sequence defined by recurrence with positive integers.	Algebra	Sequences & Series	13
2009 AIME I	Minimize S2 given S1=513, S4=4745 for 350 numbers from {1,2,3,4}.	Algebra	Systems & Algebraic Manipulation	14
2009 AIME I	Maximum area of triangle BPC with incenters and circumcircles.	Geometry	Triangles & Polygons	15
2009 AIME II	Find total paint left after painting four equal stripes with given initial amounts.	Algebra	Systems & Algebraic Manipulation	1
2009 AIME II	Evaluate sum of three exponential expressions with given logarithmic equations.	Algebra	Logarithms & Exponents	2
2009 AIME II	Find greatest integer less than AD in rectangle with perpendicular lines.	Geometry	Triangles & Polygons	3
2009 AIME II	Find smallest n such that sum of arithmetic sequence equals 2009.	Algebra	Sequences & Series	4
2009 AIME II	Find radius of circle tangent to three circles inscribed in a larger circle.	Geometry	Circular Geometry	5
2009 AIME II	Count 5-element subsets of {1..14} with at least two consecutive numbers.	Combinatorics	Counting	6
2009 AIME II	Find ab/10 where denominator of sum of double factorial fractions is 2^a * b.	Number Theory	Factors, Multiples & Divisibility	7
2009 AIME II	Probability that two people rolling for a six have roll counts within one.	Combinatorics	Probability & Expected Value	8
2009 AIME II	Find difference in number of positive integer solutions to two linear Diophantine equations.	Number Theory	Diophantine Equations	9
2009 AIME II	Find distance AD given lighthouses with equal angles and given distances.	Geometry	Triangles & Polygons	10
2009 AIME II	Find sum of all possible mn given exactly 50 integers k satisfy inequality.	Algebra	Logarithms & Exponents	11
2009 AIME II	Maximize number of disjoint pairs from 1..2009 with distinct sums ≤2009.	Combinatorics	Counting	12
2009 AIME II	Find product of 12 chords in a semicircle divided into 7 equal arcs.	Geometry	Trigonometry	13
2009 AIME II	Find greatest integer ≤ a_10 given recurrence with square root.	Algebra	Sequences & Series	14
2009 AIME II	Maximize segment length between intersections of chords with diameter in a circle.	Geometry	Circular Geometry	15
2008 AIME I	Find number of students who like to dance after 20 boys join.	Algebra	Systems & Algebraic Manipulation	1
2008 AIME I	Find altitude length of isosceles triangle given intersection area with square.	Geometry	Triangles & Polygons	2
2008 AIME I	Find sum of squares of biking, jogging, swimming rates.	Algebra	Systems & Algebraic Manipulation	3
2008 AIME I	Find sum of positive integers x and y satisfying quadratic Diophantine equation.	Number Theory	Diophantine Equations	4
2008 AIME I	Find h/r for cone rolling 17 rotations on its side.	Geometry	3D Geometry & Solids	5
2008 AIME I	Count entries divisible by 67 in triangular array of odd numbers.	Number Theory	Factors, Multiples & Divisibility	6
2008 AIME I	Count sets of 100 consecutive integers containing no perfect square.	Number Theory	Factors, Multiples & Divisibility	7
2008 AIME I	Find n such that sum of arctangents equals pi/4.	Geometry	Trigonometry	8
2008 AIME I	Find probability stack of 10 crates is exactly 41 feet tall.	Combinatorics	Probability & Expected Value	9
2008 AIME I	Find distance EF in isosceles trapezoid with given diagonals and point E.	Geometry	Triangles & Polygons	10
2008 AIME I	Count sequences of A and B of length 14 with run length constraints.	Combinatorics	Recursion in Combinatorics	11
2008 AIME I	Find maximum number of cars passing a point in one hour given safety rule.	Algebra	Systems & Algebraic Manipulation	12
2008 AIME I	Find common zero of all cubic polynomials vanishing at eight given points.	Algebra	Polynomials	13
2008 AIME I	Find maximum possible length of BP in tangent configuration with diameter 18.	Geometry	Circular Geometry	14
2008 AIME I	Find height of paper tray formed by folding cut square corners.	Geometry	3D Geometry & Solids	15
2008 AIME II	Evaluate alternating sum of squares and find remainder mod 1000.	Algebra	Sequences & Series	1
2008 AIME II	Find total time for two bikers with different speeds and breaks.	Algebra	Systems & Algebraic Manipulation	2
2008 AIME II	Maximize remaining volume after ten slices cut from a rectangular block.	Algebra	Inequalities & Optimization	3
2008 AIME II	Express 2008 as signed sum of powers of 3 and sum exponents.	Number Theory	Base Representations	4
2008 AIME II	Find distance between midpoints of bases in a trapezoid.	Geometry	Triangles & Polygons	5
2008 AIME II	Find ratio of terms in two recursively defined sequences.	Algebra	Sequences & Series	6
2008 AIME II	Find sum of cubes of pairwise sums of roots of a cubic.	Algebra	Polynomials	7
2008 AIME II	Find smallest n making a trigonometric sum an integer.	Geometry	Trigonometry	8
2008 AIME II	Find position of particle after 150 moves of rotation and translation.	Algebra	Complex Numbers	9
2008 AIME II	Count maximum-length paths with strictly increasing distances in a 4x4 grid.	Combinatorics	Counting	10
2008 AIME II	Find radius of circle tangent to two sides and another circle in a triangle.	Geometry	Triangles & Polygons	11
2008 AIME II	Count arrangements of flags on two poles with no adjacent green flags.	Combinatorics	Counting	12
2008 AIME II	Find area of image of region outside hexagon under inversion.	Geometry	Triangles & Polygons	13
2008 AIME II	Maximize a/b given a system of equations with constraints.	Algebra	Inequalities & Optimization	14
2008 AIME II	Find largest n such that n^2 is difference of consecutive cubes and 2n+79 is square.	Number Theory	Diophantine Equations	15
2007 AIME I	Count positive perfect squares less than 10^6 that are multiples of 24.	Number Theory	Factors, Multiples & Divisibility	1
2007 AIME I	Find distance from start to middle person when one is halfway between the other two.	Algebra	Systems & Algebraic Manipulation	2
2007 AIME I	Find positive real b such that imaginary parts of z^2 and z^3 are equal.	Algebra	Complex Numbers	3
2007 AIME I	Find fewest years until three planets and star are collinear again.	Number Theory	Modular Arithmetic	4
2007 AIME I	Count integer Fahrenheit temperatures where rounding conversion yields original value.	Number Theory	Modular Arithmetic	5
2007 AIME I	Count move sequences from 0 to 39 using multiples of 3 or 13.	Combinatorics	Recursion in Combinatorics	6
2007 AIME I	Find remainder when sum involving floor/ceiling of log base sqrt(2) is divided by 1000.	Algebra	Sequences & Series	7
2007 AIME I	Find largest k such that two quadratics are both factors of a cubic.	Algebra	Polynomials	8
2007 AIME I	Find radius of two equal circles tangent to hypotenuse and leg extensions.	Geometry	Triangles & Polygons	9
2007 AIME I	Count shadings of 6x4 grid with 2 shaded per row and 3 per column.	Combinatorics	Counting	10
2007 AIME I	Sum over p of integer nearest sqrt(p) for p=1 to 2007, mod 1000.	Number Theory	Modular Arithmetic	11
2007 AIME I	Find area common to original and rotated isosceles triangle, then compute expression.	Geometry	Trigonometry	12
2007 AIME I	Find area of plane intersection with square pyramid, expressed as sqrt(p).	Geometry	3D Geometry & Solids	13
2007 AIME I	Find greatest integer ≤ (a_2006^2 + a_2007^2)/(a_2006 a_2007) for given recurrence.	Algebra	Sequences & Series	14
2007 AIME I	Find r in side length expression p ± q√r for equilateral triangle with interior 60° angle.	Geometry	Triangles & Polygons	15
2007 AIME II	Count license plates with characters from AIME and 2007.	Combinatorics	Counting	1
2007 AIME II	Count ordered triples (a,b,c) with a\|b, a\|c, a+b+c=100.	Number Theory	Factors, Multiples & Divisibility	2
2007 AIME II	Find EF^2 in a square with external right triangles.	Geometry	Triangles & Polygons	3
2007 AIME II	Find m given production rates of widgets and whoosits.	Algebra	Systems & Algebraic Manipulation	4
2007 AIME II	Count unit squares below line 9x+223y=2007 in first quadrant.	Number Theory	Modular Arithmetic	5
2007 AIME II	Count four-digit parity-monotonic integers.	Combinatorics	Counting	6
2007 AIME II	Find max n_i/k given floor cube root and divisibility conditions.	Number Theory	Factors, Multiples & Divisibility	7
2007 AIME II	Maximize basic rectangles from lines on a 4x5 paper.	Algebra	Inequalities & Optimization	8
2007 AIME II	Find PQ where incircles of two triangles touch EF.	Geometry	Triangles & Polygons	9
2007 AIME II	Probability that B is subset of A or complement of A.	Combinatorics	Probability & Expected Value	10
2007 AIME II	Distance large tube rolls over small tube without slipping.	Geometry	Circular Geometry	11
2007 AIME II	Find log_3(x_14) given sum of logs and sum bounds.	Algebra	Sequences & Series	12
2007 AIME II	Count bottom row 0/1 distributions making top sum multiple of 3.	Combinatorics	Counting	13
2007 AIME II	Find f(5) given functional equation and values.	Algebra	Polynomials	14
2007 AIME II	Find radius of circle tangent to three equal circles in triangle.	Geometry	Circular Geometry	15
2006 AIME I	Find the perimeter of quadrilateral ABCD with right angles and perpendicular diagonals.	Geometry	Triangles & Polygons	1
2006 AIME I	Find the number of possible sums of a 90-element subset of 1 to 100.	Combinatorics	Counting	2
2006 AIME I	Find the least positive integer such that deleting its leftmost digit yields 1/29 of it.	Number Theory	Base Representations	3
2006 AIME I	Find the remainder when the number of trailing zeros of 1!2!...100! is divided by 1000.	Number Theory	Factors, Multiples & Divisibility	4
2006 AIME I	Find abc where sqrt(104√6+468√10+144√15+2006) = a√2+b√3+c√5.	Algebra	Systems & Algebraic Manipulation	5
2006 AIME I	Find the sum of all numbers 0.abc with distinct digits a,b,c.	Combinatorics	Counting	6
2006 AIME I	Find ratio of area D to area A given ratio of C to B is 11/5.	Geometry	Coordinate Geometry	7
2006 AIME I	Find number of possible integer areas K of rhombus T given four congruent rhombi.	Geometry	Triangles & Polygons	8
2006 AIME I	Find number of ordered pairs (a,r) given log_8 sum of geometric sequence equals 2006.	Algebra	Logarithms & Exponents	9
2006 AIME I	Find a^2+b^2+c^2 for line with slope 3 dividing eight circles into equal area.	Geometry	Coordinate Geometry	10
2006 AIME I	Find remainder when number of towers from 8 cubes with edge-length rule is divided by 1000.	Combinatorics	Recursion in Combinatorics	11
2006 AIME I	Find sum of x in (100,200) satisfying cos^3 3x+cos^3 5x=8 cos^3 4x cos^3 x.	Geometry	Trigonometry	12
2006 AIME I	Find greatest n<1000 such that sum of greatest powers of 2 dividing even numbers is a perfect square.	Number Theory	Factors, Multiples & Divisibility	13
2006 AIME I	Find floor(m+√n) for height of broken tripod with legs of length 5.	Geometry	3D Geometry & Solids	14
2006 AIME I	Find minimum possible absolute sum of sequence with \|x_k\|=\|x_{k-1}+3\| and x_0=0.	Algebra	Sequences & Series	15
2006 AIME II	Find side length of convex hexagon with congruent sides and right angles.	Geometry	Triangles & Polygons	1
2006 AIME II	Count integer n such that logs of 12, 75, n form triangle.	Algebra	Logarithms & Exponents	2
2006 AIME II	Find largest k such that product of first 100 odd numbers divisible by 3^k.	Number Theory	Factors, Multiples & Divisibility	3
2006 AIME II	Count permutations of 1..12 with decreasing then increasing sequence.	Combinatorics	Counting	4
2006 AIME II	Find probability of face F on unfair die given sum 7 probability.	Combinatorics	Probability & Expected Value	5
2006 AIME II	Find side length of square inside unit square with equilateral triangle.	Geometry	Triangles & Polygons	6
2006 AIME II	Count ordered pairs (a,b) with a+b=1000 and neither has zero digit.	Combinatorics	Counting	7
2006 AIME II	Count distinguishable colorings of large triangle made of four small triangles.	Combinatorics	Counting	8
2006 AIME II	Find x-coordinate of intersection of two common internal tangents to circles.	Geometry	Coordinate Geometry	9
2006 AIME II	Probability team A finishes with more points than B after first game win.	Combinatorics	Probability & Expected Value	10
2006 AIME II	Find remainder of sum of first 28 terms of tribonacci-like sequence mod 1000.	Algebra	Sequences & Series	11
2006 AIME II	Find area of triangle CBG in circle with equilateral triangle and parallels.	Geometry	Triangles & Polygons	12
2006 AIME II	Count integers less than 1000 expressible as sum of j consecutive odd integers in exactly 5 ways.	Number Theory	Factors, Multiples & Divisibility	13
2006 AIME II	Find smallest n such that sum of reciprocals of non-zero digits from 1 to 10^n is integer.	Number Theory	Base Representations	14
2006 AIME II	Find x+y+z given three equations involving square roots of squares minus constants.	Algebra	Systems & Algebraic Manipulation	15
2005 AIME I	Six congruent circles in a ring inside a larger circle; find floor of area between them.	Geometry	Circular Geometry	1
2005 AIME I	Count values of k such that arithmetic sequence S_k contains 2005.	Number Theory	Factors, Multiples & Divisibility	2
2005 AIME I	Count positive integers with exactly three proper divisors each less than 50.	Number Theory	Factors, Multiples & Divisibility	3
2005 AIME I	Find maximum band members given square formation leaves 5, rectangular formation has 7 more rows than columns.	Number Theory	Diophantine Equations	4
2005 AIME I	Count distinguishable stacks of 4 gold and 4 silver coins with no adjacent faces touching.	Combinatorics	Counting	5
2005 AIME I	Find floor of product of nonreal roots of x^4-4x^3+6x^2-4x=2005.	Algebra	Polynomials	6
2005 AIME I	Find AB in quadrilateral with given sides and two 60-degree angles.	Geometry	Triangles & Polygons	7
2005 AIME I	Find sum of three real roots of exponential equation in terms of m/n.	Algebra	Polynomials	8
2005 AIME I	Probability that entire surface of 3x3x3 cube is orange after random arrangement.	Combinatorics	Probability & Expected Value	9
2005 AIME I	Find largest possible p+q for vertex A of triangle with given area and median slope.	Geometry	Coordinate Geometry	10
2005 AIME I	Find maximum diameter of semicircle inside an 8x8 square.	Geometry	Circular Geometry	11
2005 AIME I	Find \|a-b\| where a counts n with odd S(n) and b counts n with even S(n) up to 2005.	Number Theory	Factors, Multiples & Divisibility	12
2005 AIME I	Count paths from (0,0) to (5,5) with moves right, up, or diagonal and no right-angle turns.	Combinatorics	Counting	13
2005 AIME I	Find area of square with each of four given points on a different side.	Geometry	Coordinate Geometry	14
2005 AIME I	Find area of triangle with BC=20 and incircle trisecting median AD.	Geometry	Triangles & Polygons	15
2005 AIME II	Find n such that C(n,6) = 6 * C(n,3).	Combinatorics	Counting	1
2005 AIME II	Probability each of three guests gets one of each roll type.	Combinatorics	Probability & Expected Value	2
2005 AIME II	Infinite geometric series with sum 2005; squaring each term gives series with 10x the sum. Common ratio m/n (relatively prime); find m+n.	Algebra	Sequences & Series	3
2005 AIME II	Count divisors of at least one of 10^10, 15^7, 18^11.	Number Theory	Factors, Multiples & Divisibility	4
2005 AIME II	Count ordered pairs (a,b) satisfying log_a b + 6 log_b a = 5.	Algebra	Logarithms & Exponents	5
2005 AIME II	Find number of cards where card 131 retains original position.	Combinatorics	Counting	6
2005 AIME II	Simplify expression involving nested radicals of 5, find (x+1)^48.	Algebra	Systems & Algebraic Manipulation	7
2005 AIME II	Find chord length of circle C3 tangent to C1 and C2.	Geometry	Circular Geometry	8
2005 AIME II	Count n ≤ 1000 where (sin t + i cos t)^n = sin nt + i cos nt.	Algebra	Complex Numbers	9
2005 AIME II	Ratio of volume of regular octahedron to its dual cube.	Geometry	3D Geometry & Solids	10
2005 AIME II	Find m given recurrence a_{k+1} = a_{k-1} - 3/a_k with a_0=37, a_1=72, a_m=0.	Algebra	Sequences & Series	11
2005 AIME II	Find BF in square ABCD with given lengths and 45° angle.	Geometry	Triangles & Polygons	12
2005 AIME II	Find product of integer solutions n to P(n)=n+3 given P(17)=10, P(24)=17.	Algebra	Polynomials	13
2005 AIME II	Find BE in triangle ABC with given side lengths and equal angles.	Geometry	Triangles & Polygons	14
2005 AIME II	Find smallest positive a so line y=ax contains center of circle tangent to two given circles.	Geometry	Coordinate Geometry	15
2004 AIME I	Sum of remainders when four-digit numbers with consecutive decreasing digits are divided by 37.	Number Theory	Modular Arithmetic	1
2004 AIME I	Find m given sums of consecutive integers in two sets and difference of largest elements.	Algebra	Systems & Algebraic Manipulation	2
2004 AIME I	Count space diagonals of a convex polyhedron given vertices, edges, and faces.	Combinatorics	Counting	3
2004 AIME I	Area enclosed by midpoints of length-2 segments on adjacent sides of a square.	Geometry	Coordinate Geometry	4
2004 AIME I	Largest possible two-day success ratio for Beta given constraints on daily scores.	Algebra	Inequalities & Optimization	5
2004 AIME I	Count four-digit snakelike numbers with distinct digits between 1000 and 9999.	Combinatorics	Counting	6
2004 AIME I	Find absolute value of coefficient of x^2 in product (1-x)(1+2x)...(1-15x).	Algebra	Polynomials	7
2004 AIME I	Number of non-similar regular 1000-pointed stars.	Number Theory	Modular Arithmetic	8
2004 AIME I	Minimum area of triangle similar to 3-4-5 triangle cut from rectangle.	Geometry	Triangles & Polygons	9
2004 AIME I	Probability that a unit circle placed in a 15-by-36 rectangle does not touch diagonal.	Geometry	Coordinate Geometry	10
2004 AIME I	Find k where painted surface area ratio equals volume ratio for cone cut by parallel plane.	Geometry	3D Geometry & Solids	11
2004 AIME I	Area of set where floor(log2(1/x)) and floor(log5(1/y)) are both even.	Algebra	Functions	12
2004 AIME I	Sum of five smallest arguments of complex roots of polynomial P(x).	Algebra	Complex Numbers	13
2004 AIME I	Length of rope touching cylindrical tower when unicorn pulls it taut.	Geometry	3D Geometry & Solids	14
2004 AIME I	Number of positive integers x such that d(x)=20 under given function f.	Combinatorics	Recursion in Combinatorics	15
2004 AIME II	Ratio of areas formed by a chord perpendicular to a radius at its midpoint.	Geometry	Circular Geometry	1
2004 AIME II	Probability Terry and Mary pick same color combination of candies.	Combinatorics	Probability & Expected Value	2
2004 AIME II	Minimize number of 1-cm cubes in a rectangular block with 231 invisible cubes.	Number Theory	Factors, Multiples & Divisibility	3
2004 AIME II	Count positive integers less than 10000 with at most two different digits.	Combinatorics	Counting	4
2004 AIME II	Minimum additional workers needed to complete a job on schedule after layoffs.	Algebra	Systems & Algebraic Manipulation	5
2004 AIME II	Least total bananas given ratios after three monkeys take and share.	Algebra	Systems & Algebraic Manipulation	6
2004 AIME II	Perimeter of rectangle after folding corner B to point B' on AD.	Geometry	Triangles & Polygons	7
2004 AIME II	Number of divisors of 2004^2004 divisible by exactly 2004 positive integers.	Number Theory	Factors, Multiples & Divisibility	8
2004 AIME II	Find n + a_n for greatest term less than 1000 in a mixed progression sequence.	Algebra	Sequences & Series	9
2004 AIME II	Probability a number with two 1's in binary up to 2^40 is divisible by 9.	Number Theory	Modular Arithmetic	10
2004 AIME II	Shortest distance a fly crawls on a cone surface between two points.	Geometry	3D Geometry & Solids	11
2004 AIME II	Radius of circle tangent to four circles centered at trapezoid vertices.	Geometry	Circular Geometry	12
2004 AIME II	Ratio of areas of triangles ABC and EBD in a convex pentagon with parallels.	Geometry	Triangles & Polygons	13
2004 AIME II	Number of n such that inserting + into n 7's yields 7000.	Number Theory	Diophantine Equations	14
2004 AIME II	Squares below the original 942nd square after 10 folds of a 1024 strip.	Combinatorics	Recursion in Combinatorics	15
2003 AIME I	Simplify factorial expression to find largest possible n and k+n.	Algebra	Systems & Algebraic Manipulation	1
2003 AIME I	Find ratio of green area to largest circle in concentric circles.	Algebra	Sequences & Series	2
2003 AIME I	Sum of larger element in each two-element subset of a set.	Combinatorics	Counting	3
2003 AIME I	Solve logarithmic equations involving sin and cos to find n.	Algebra	Logarithms & Exponents	4
2003 AIME I	Volume of points within one unit of a 3x4x5 box.	Geometry	3D Geometry & Solids	5
2003 AIME I	Sum of areas of all triangles with vertices of a unit cube.	Geometry	3D Geometry & Solids	6
2003 AIME I	Find sum of perimeters of isosceles triangle with integer sides.	Number Theory	Diophantine Equations	7
2003 AIME I	Find sum of four terms in arithmetic and geometric progression.	Algebra	Sequences & Series	8
2003 AIME I	Count 4-digit numbers where sum of first two digits equals last two.	Combinatorics	Counting	9
2003 AIME I	Find angle CMB in isosceles triangle with given interior angles.	Geometry	Triangles & Polygons	10
2003 AIME I	Probability sin^2 x, cos^2 x, sin x cos x do not form a triangle.	Geometry	Trigonometry	11
2003 AIME I	Find floor of 1000 cos A in quadrilateral with equal angles and sides.	Geometry	Triangles & Polygons	12
2003 AIME I	Count numbers ≤2003 with more 1s than 0s in binary, mod 1000.	Combinatorics	Counting	13
2003 AIME I	Find smallest denominator n so decimal of m/n contains 251 consecutively.	Number Theory	Base Representations	14
2003 AIME I	Find DE:EF ratio in triangle with angle bisector and median.	Geometry	Triangles & Polygons	15
2003 AIME II	Find sum of all possible products of three positive integers with given conditions.	Algebra	Systems & Algebraic Manipulation	1
2003 AIME II	Find remainder when greatest integer multiple of 8 with distinct digits is divided by 1000.	Number Theory	Modular Arithmetic	2
2003 AIME II	Count seven-letter words from A, B, C with no forbidden adjacent pairs.	Combinatorics	Counting	3
2003 AIME II	Find ratio of volumes of tetrahedron formed by face centers to original tetrahedron.	Geometry	3D Geometry & Solids	4
2003 AIME II	Find volume of wedge cut from a cylindrical log by two planar cuts.	Geometry	3D Geometry & Solids	5
2003 AIME II	Find area of union of triangle and its 180-degree rotation about centroid.	Geometry	Triangles & Polygons	6
2003 AIME II	Find area of rhombus given circumradii of two triangles formed by its diagonals.	Geometry	Triangles & Polygons	7
2003 AIME II	Find eighth term of sequence formed by multiplying corresponding terms of two arithmetic sequences.	Algebra	Sequences & Series	8
2003 AIME II	Find sum of P evaluated at roots of Q, where P and Q are given polynomials.	Algebra	Polynomials	9
2003 AIME II	Find maximum possible sum of two positive integers differing by 60 with given square root condition.	Number Theory	Diophantine Equations	10
2003 AIME II	Find area of triangle CDM in right triangle ABC with given side lengths and point D.	Geometry	Triangles & Polygons	11
2003 AIME II	Find smallest possible number of committee members given vote percentage condition.	Number Theory	Inequalities & Optimization	12
2003 AIME II	Find probability bug returns to starting vertex after ten moves on equilateral triangle.	Combinatorics	Recursion in Combinatorics	13
2003 AIME II	Find area of convex equilateral hexagon with given angle and parallel side conditions.	Geometry	Coordinate Geometry	14
2003 AIME II	Find sum of absolute imaginary parts of squares of distinct zeros of given polynomial.	Algebra	Polynomials	15
2002 AIME I	Probability a license plate has a three-letter or three-digit palindrome.	Combinatorics	Probability & Expected Value	1
2002 AIME I	Ratio of rectangle dimensions given tangent circles.	Geometry	Circular Geometry	2
2002 AIME I	Number of ordered pairs (Dick's age, years) with digit reversal.	Number Theory	Base Representations	3
2002 AIME I	Find m+n given telescoping sum equals 1/29.	Algebra	Sequences & Series	4
2002 AIME I	Number of squares with at least two vertices of a regular dodecagon.	Combinatorics	Counting	5
2002 AIME I	Evaluate log base 30 of product of solutions to log system.	Algebra	Logarithms & Exponents	6
2002 AIME I	First three decimal digits of (10^2002+1)^(10/7).	Number Theory	Modular Arithmetic	7
2002 AIME I	Smallest k such that more than one Fibonacci-like sequence exists.	Algebra	Sequences & Series	8
2002 AIME I	Sum of paintable integers 100h+10t+u for fence painting.	Number Theory	Modular Arithmetic	9
2002 AIME I	Area of quadrilateral DCFG in right triangle with angle bisector.	Geometry	Triangles & Polygons	10
2002 AIME I	Length of light path reflecting off cube faces until vertex.	Geometry	3D Geometry & Solids	11
2002 AIME I	Find a+b for complex iteration z_n = F(z_{n-1}) with period 3.	Algebra	Complex Numbers	12
2002 AIME I	Area of triangle AFB with medians and circumcircle.	Geometry	Triangles & Polygons	13
2002 AIME I	Maximum size of set S with integer mean after deleting any element.	Number Theory	Modular Arithmetic	14
2002 AIME I	Compute EG^2 for polyhedron with square, trapezoid, and triangle faces.	Geometry	3D Geometry & Solids	15
2002 AIME II	Count distinct absolute differences between three-digit numbers and their reversals.	Number Theory	Base Representations	1
2002 AIME II	Find surface area of a cube given three vertices.	Geometry	3D Geometry & Solids	2
2002 AIME II	Find sum of three positive integers in geometric sequence with given log condition.	Algebra	Logarithms & Exponents	3
2002 AIME II	Find area of garden enclosed by hexagonal blocks with n=202.	Algebra	Sequences & Series	4
2002 AIME II	Sum positive integers of form 2^n 3^m where expression is not a divisor.	Number Theory	Factors, Multiples & Divisibility	5
2002 AIME II	Find integer closest to 1000 times sum of 1/(n^2-4) from n=3 to 10000.	Algebra	Sequences & Series	6
2002 AIME II	Find smallest positive k such that sum of squares is multiple of 200.	Number Theory	Modular Arithmetic	7
2002 AIME II	Find least positive integer k with no integer n satisfying floor(2002/n)=k.	Number Theory	Modular Arithmetic	8
2002 AIME II	Count unordered pairs of non-empty disjoint subsets of {1,...,10}.	Combinatorics	Counting	9
2002 AIME II	Find sum m+n+p+q for two least positive x with sin(x deg)=sin(x rad).	Geometry	Trigonometry	10
2002 AIME II	Find second term of two geometric series with sum 1 and same second term.	Algebra	Sequences & Series	11
2002 AIME II	Probability shooting average is 0.4 after 10 shots and never exceeds 0.4.	Combinatorics	Probability & Expected Value	12
2002 AIME II	Find ratio of area of triangle PQR to area of triangle ABC.	Geometry	Triangles & Polygons	13
2002 AIME II	Find OP given triangle perimeter, right angle, and inscribed circle tangent to sides.	Geometry	Triangles & Polygons	14
2002 AIME II	Find m in y=mx tangent to two circles intersecting at (9,6) with radii product 68.	Geometry	Coordinate Geometry	15
2001 AIME I	Sum of two-digit integers divisible by each of their digits	Number Theory	Factors, Multiples & Divisibility	1
2001 AIME I	Mean of a set given means after adding 1 and 2001	Algebra	Systems & Algebraic Manipulation	2
2001 AIME I	Sum of all roots of x^2001 + (1/2 - x)^2001 = 0	Algebra	Polynomials	3
2001 AIME I	Area of triangle with given angles and angle bisector length	Geometry	Triangles & Polygons	4
2001 AIME I	Side length of equilateral triangle inscribed in ellipse	Geometry	Coordinate Geometry	5
2001 AIME I	Probability that four die rolls are non-decreasing	Combinatorics	Probability & Expected Value	6
2001 AIME I	Length of segment through incenter parallel to base	Geometry	Triangles & Polygons	7
2001 AIME I	Largest integer whose base-7 digits equal twice the number in base-10	Number Theory	Base Representations	8
2001 AIME I	Ratio of areas of triangle DEF to ABC given side ratios	Geometry	Triangles & Polygons	9
2001 AIME I	Probability that midpoint of two lattice points is also a lattice point	Combinatorics	Probability & Expected Value	10
2001 AIME I	Smallest N for consistent renumbering of a 5xN array	Number Theory	Modular Arithmetic	11
2001 AIME I	Inradius of tetrahedron with vertices at (6,0,0), (0,4,0), (0,0,2), (0,0,0)	Geometry	3D Geometry & Solids	12
2001 AIME I	Chord length of 3d-degree arc given chord lengths of d and 2d arcs	Geometry	Trigonometry	13
2001 AIME I	Number of 19-house mail patterns with no two adjacent and no three empty	Combinatorics	Recursion in Combinatorics	14
2001 AIME I	Probability no consecutive numbers on adjacent octahedron faces	Combinatorics	Counting	15
2001 AIME II	Largest integer where each consecutive digit pair forms a perfect square.	Number Theory	Base Representations	1
2001 AIME II	Find difference between max and min students studying both languages.	Combinatorics	Counting	2
2001 AIME II	Sum of three terms of a recursively defined sequence.	Algebra	Sequences & Series	3
2001 AIME II	Length of segment PQ with R as midpoint on given lines.	Geometry	Coordinate Geometry	4
2001 AIME II	Largest n so all ten-element subsets of {4,...,n} have triangle property.	Combinatorics	Counting	5
2001 AIME II	Area of square inscribed in circle with another square on its side.	Geometry	Circular Geometry	6
2001 AIME II	Distance between centers of incircles of two triangles in a right triangle.	Geometry	Triangles & Polygons	7
2001 AIME II	Smallest x such that f(x)=f(2001) given functional equation and piecewise definition.	Algebra	Functions	8
2001 AIME II	Probability a 3x3 grid has no 2x2 red square when each cell is colored randomly.	Combinatorics	Probability & Expected Value	9
2001 AIME II	Number of positive multiples of 1001 of the form 10^j - 10^i.	Number Theory	Modular Arithmetic	10
2001 AIME II	Probability soccer team has more wins than losses in 6 games with equal win/loss/tie odds.	Combinatorics	Probability & Expected Value	11
2001 AIME II	Volume after recursively adding tetrahedra to faces of a regular tetrahedron.	Geometry	3D Geometry & Solids	12
2001 AIME II	Length CD in quadrilateral with given angle equalities and side lengths.	Geometry	Triangles & Polygons	13
2001 AIME II	Sum of even-indexed arguments of complex numbers on unit circle satisfying z^28 - z^8 - 1 = 0.	Algebra	Complex Numbers	14
2001 AIME II	Surface area of cube with triangular tunnel drilled through it.	Geometry	3D Geometry & Solids	15
2000 AIME I	Least n such that any factorization of 10^n has a factor with digit 0.	Number Theory	Factors, Multiples & Divisibility	1
2000 AIME I	Area of pentagon formed by reflections of (u,v) is 451; find u+v.	Geometry	Coordinate Geometry	2
2000 AIME I	Coefficients of x^2 and x^3 in (ax+b)^2000 equal; find a+b.	Algebra	Polynomials	3
2000 AIME I	Rectangle dissected into nine squares; find perimeter given relatively prime sides.	Algebra	Systems & Algebraic Manipulation	4
2000 AIME I	Probability both marbles white given total 25 and black probability 27/50.	Combinatorics	Probability & Expected Value	5
2000 AIME I	Number of integer pairs (x,y) with 0<x<y<10^6 and arithmetic mean 2 more than geometric mean.	Algebra	Systems & Algebraic Manipulation	6
2000 AIME I	Given xyz=1, x+1/z=5, y+1/x=29, find z+1/y as reduced fraction.	Algebra	Systems & Algebraic Manipulation	7
2000 AIME I	Height of liquid in inverted cone given original depth 9 inches.	Geometry	3D Geometry & Solids	8
2000 AIME I	System of log equations with two solutions; find sum of y-values.	Algebra	Logarithms & Exponents	9
2000 AIME I	Sequence where x_k is k less than sum of other 99 terms; find x_50.	Algebra	Sequences & Series	10
2000 AIME I	Sum of all a/b with a,b relatively prime divisors of 1000; floor of S/10.	Number Theory	Factors, Multiples & Divisibility	11
2000 AIME I	Function symmetric about three lines; max distinct values from 0 to 999.	Algebra	Functions	12
2000 AIME I	Area reachable by firetruck in 6 minutes on highways and prairie.	Geometry	Coordinate Geometry	13
2000 AIME I	Isosceles triangle with AP=PQ=QB=BC; find floor(1000 * angle ratio).	Geometry	Triangles & Polygons	14
2000 AIME I	Card shuffling process yields ascending order; find cards above 1999 originally.	Combinatorics	Recursion in Combinatorics	15
2000 AIME II	Simplify sum of fractions with logarithms to find m+n.	Algebra	Logarithms & Exponents	1
2000 AIME II	Count lattice points on hyperbola x^2 - y^2 = 2000^2.	Number Theory	Diophantine Equations	2
2000 AIME II	Probability two cards form a pair after one pair removed.	Combinatorics	Probability & Expected Value	3
2000 AIME II	Find smallest integer with 6 odd and 12 even divisors.	Number Theory	Factors, Multiples & Divisibility	4
2000 AIME II	Count arrangements of rings on fingers with order significance.	Combinatorics	Counting	5
2000 AIME II	Find x^2/100 floor for trapezoid with equal area division.	Geometry	Triangles & Polygons	6
2000 AIME II	Evaluate sum of reciprocals of factorials to find N/100 floor.	Combinatorics	Counting	7
2000 AIME II	Find BC^2 in trapezoid with perpendicular diagonals.	Geometry	Triangles & Polygons	8
2000 AIME II	Find least integer greater than \|z\| for complex z satisfying equation.	Algebra	Complex Numbers	9
2000 AIME II	Find square of radius of circle inscribed in quadrilateral.	Geometry	Circular Geometry	10
2000 AIME II	Sum absolute slopes of AB in isosceles trapezoid with integer vertices.	Geometry	Coordinate Geometry	11
2000 AIME II	Find distance from sphere center to triangle ABC on its surface.	Geometry	3D Geometry & Solids	12
2000 AIME II	Find m+n+r for real root of polynomial expressed as (m+√n)/r.	Algebra	Polynomials	13
2000 AIME II	Evaluate alternating sum of factorial base digits of large expression.	Number Theory	Base Representations	14
2000 AIME II	Find least n such that sum of cosecant products equals 1/sin n°.	Geometry	Trigonometry	15

2025 AIME I #1

Modular Arithmetic▼

Find sum of integer bases b>9 where 17_b divides 97_b.

2025 AIME I #2

Triangles & Polygons▼

Find area of heptagon formed by reflections in triangle with given side ratios.

2025 AIME I #3

Counting▼

Count assignments of 9 players to 3 ice cream flavors with ordering constraints.

2025 AIME I #4

Systems & Algebraic Manipulation▼

Count integer pairs (x,y) in range satisfying 12x^2 - xy - 6y^2 = 0.

2025 AIME I #5

Counting▼

Count 8-digit permutations of 1-8 divisible by 22, subtract 2025.

2025 AIME I #6

Triangles & Polygons▼

Find r^2+s^2 for isosceles trapezoid with inscribed circle radius 3 and area 72.

2025 AIME I #7

Probability & Expected Value▼

Probability last word contains G when pairing 12 letters into 6 words.

2025 AIME I #8

Complex Numbers▼

Find sum of k values for which complex system has exactly one solution.

2025 AIME I #9

Functions▼

Find y-coordinate of intersection of parabola and its 60-degree rotation.

2025 AIME I #10

Counting▼

Count ways to fill 3x9 Sudoku-like grid with numbers 1-9.

2025 AIME I #11

Functions▼

Sum y-coordinates of intersections of sawtooth function and parabola x=34y^2.

2025 AIME I #12

Inequalities & Optimization▼

Find area of finite region in plane x+y+z=75 satisfying inequalities.

2025 AIME I #13

Probability & Expected Value▼

Expected number of regions when 27 segments divide a disk.

2025 AIME I #14

Triangles & Polygons▼

Minimize sum of distances from point X to vertices of convex pentagon.

2025 AIME I #15

Factors, Multiples & Divisibility▼

Count triples (a,b,c) up to 3^6 with a^3+b^3+c^3 divisible by 3^7.

2025 AIME II #1

Triangles & Polygons▼

Find area of triangle BGE given collinear points and distances.

2025 AIME II #2

Modular Arithmetic▼

Sum of positive n such that n+2 divides 3(n+3)(n^2+9).

2025 AIME II #3

Counting▼

Number of colorings of 2x2 grid edges with 2 red and 2 blue per square.

2025 AIME II #4

Logarithms & Exponents▼

Evaluate product of logarithms and find sum of numerator and denominator.

2025 AIME II #5

Triangles & Polygons▼

Find sum of arcs on circumcircle of medial triangle given triangle angles.

2025 AIME II #6

Circular Geometry▼

Find area of rectangle inscribed in circle tangent to larger circle.

2025 AIME II #7

Probability & Expected Value▼

Probability that LCM of random subset of divisors of 2025 is 2025.

2025 AIME II #8

▼

Count N up to 1000 where greedy coin algorithm uses fewest coins.

2025 AIME II #9

Functions▼

Find n+t where n zeros of f(x)=sin(7π sin(5x)) and t tangent zeros.

2025 AIME II #10

Counting▼

Number of subsets of 16 chairs with 8 people, no three adjacent.

2025 AIME II #11

Counting▼

Number of ways to draw 12 equal segments pairing vertices of 24-gon.

2025 AIME II #12

Trigonometry▼

Find A1A2 + A1A11 for 11-gon with given area and cosine conditions.

2025 AIME II #13

Sequences & Series▼

Find m+n mod 1000 for x_2025 = m/n from given recurrence.

2025 AIME II #14

Triangles & Polygons▼

Find n where area of quadrilateral BKLC equals n√3 in right triangle.

2025 AIME II #15

Polynomials▼

Sum of k such that f(x) minimum occurs at exactly two positive x.

2024 AIME I #1

Systems & Algebraic Manipulation▼

Find walk time including coffee break at new speed.

2024 AIME I #2

Logarithms & Exponents▼

Find product xy given two logarithmic equations.

2024 AIME I #3

Counting▼

Count n ≤ 2024 where Bob wins a token removal game.

2024 AIME I #4

Probability & Expected Value▼

Find conditional probability of winning grand prize given a prize.

2024 AIME I #5

Circular Geometry▼

Find length CE in configuration of two rectangles and a circle.

2024 AIME I #6

Counting▼

Count grid paths of length 16 with exactly four direction changes.

2024 AIME I #7

Complex Numbers▼

Maximize real part of complex expression with |z|=4.

2024 AIME I #8

Triangles & Polygons▼

Find inradius of triangle given two sequences of tangent circles.

2024 AIME I #9

Coordinate Geometry▼

Find greatest lower bound for BD^2 in rhombus on hyperbola.

2024 AIME I #10

Triangles & Polygons▼

Find length AP where AD meets circumcircle again.

2024 AIME I #11

Counting▼

Probability octagon can be rotated so blues go to former red positions.

2024 AIME I #12

Functions▼

Find number of intersections of two graphs involving absolute values and trig.

2024 AIME I #13

Modular Arithmetic▼

Find smallest m such that p^2 divides m^4+1 for least prime p.

2024 AIME I #14

3D Geometry & Solids▼

Find inradius of tetrahedron with given edge lengths.

2024 AIME I #15

Inequalities & Optimization▼

Find r^2 for smallest sphere containing all boxes with given surface area and volume.

2024 AIME II #1

Counting▼

Residents own items; find number owning all four.

2024 AIME II #2

Counting▼

List of positive integers with given sum, mode, median; find sum of squares.

2024 AIME II #3

Counting▼

2x3 grid digits; row sums to 999, column sums to 99.

2024 AIME II #4

Logarithms & Exponents▼

System of log equations; find absolute value of log expression.

2024 AIME II #5

Triangles & Polygons▼

Equilateral hexagon with parallel opposite sides; find side length.

2024 AIME II #6

Counting▼

Sets of positive integers with max in A; find sum of A.

2024 AIME II #7

Modular Arithmetic▼

Four-digit number; changing any digit to 1 gives multiple of 7.

2024 AIME II #8

3D Geometry & Solids▼

Torus inside and outside sphere; find difference of radii of tangency circles.

2024 AIME II #9

Counting▼

Place chips in 5x5 grid with row/column color constraints.

2024 AIME II #10

Triangles & Polygons▼

Triangle with incenter I, circumcenter O, IA perpendicular OI; find AB*AC.

2024 AIME II #11

Polynomials▼

Nonnegative integers a,b,c sum to 300; find number of triples satisfying equation.

2024 AIME II #12

Coordinate Geometry▼

Family of unit segments in first quadrant; find unique point C on AB.

2024 AIME II #13

Complex Numbers▼

Product over 13th roots of unity; find remainder modulo 1000.

2024 AIME II #14

Base Representations▼

Base-b two-digit numbers where digit sum equals sqrt of number.

2024 AIME II #15

Counting▼

Rectangles formed by sides/diagonals of a regular dodecagon.

2023 AIME I #1

Probability & Expected Value▼

Probability that each man stands opposite a woman in a circle

2023 AIME I #2

Logarithms & Exponents▼

Solve system of log equations for rational n

2023 AIME I #3

Counting▼

Count intersection points of 40 lines with given multiplicities

2023 AIME I #4

Factors, Multiples & Divisibility▼

Sum of m such that 13!/m is a perfect square

2023 AIME I #5

Triangles & Polygons▼

Area of square given products of distances from point on circumcircle

2023 AIME I #6

Probability & Expected Value▼

Expected number of correct guesses in a card guessing game

2023 AIME I #7

Modular Arithmetic▼

Count integers with distinct remainders modulo 2 through 6

2023 AIME I #8

Triangles & Polygons▼

Perimeter of rhombus given distances from point on incircle to sides

2023 AIME I #9

Polynomials▼

Count cubic polynomials with unique integer m not 2 satisfying p(m)=p(2)

2023 AIME I #10

Modular Arithmetic▼

Find a and sum of floor terms between -1000 and 1000

2023 AIME I #11

Counting▼

Number of subsets of {1..10} with exactly one pair of consecutive integers

2023 AIME I #12

Triangles & Polygons▼

Find tan^2 of equal angles in equilateral triangle with given side divisions

2023 AIME I #13

3D Geometry & Solids▼

Ratio of volumes of two noncongruent parallelepipeds with rhombus faces

2023 AIME I #14

Counting▼

Number of sequences of 144 hand movements visiting all positions exactly once

2023 AIME I #15

Complex Numbers▼

Largest prime p<1000 with integer complex z satisfying triangle condition

2023 AIME II #1

Sequences & Series▼

Six apple tree counts form an arithmetic sequence; greatest is double least; sum is 990.

2023 AIME II #2

Base Representations▼

Find greatest integer < 1000 that is a palindrome in base 10 and base 8.

2023 AIME II #3

Triangles & Polygons▼

Isosceles right triangle ABC with interior point P; AP=10; find area of triangle.

2023 AIME II #4

Systems & Algebraic Manipulation▼

Real numbers x,y,z satisfy three symmetric equations; find sum of squares of possible x.

2023 AIME II #5

Factors, Multiples & Divisibility▼

Positive rational r such that r and 55r have same numerator+denominator sum in lowest terms.

2023 AIME II #6

Probability & Expected Value▼

Two points chosen uniformly in L-shaped region of three unit squares; probability midpoint is inside.

2023 AIME II #7

Counting▼

Color vertices of regular dodecagon red/blue; no four same-colored vertices form a rectangle.

2023 AIME II #8

Complex Numbers▼

Product over k=0..6 of (omega^{3k}+omega^k+1) where omega is primitive 7th root of unity.

2023 AIME II #9

Circular Geometry▼

Two intersecting circles with common tangent; parallel line through P; find area of trapezoid XABY.

2023 AIME II #10

Counting▼

Place numbers 1-12 in 2x6 grid so adjacent cells differ by not divisible by 3.

2023 AIME II #11

Counting▼

Number of collections of 16 distinct subsets of {1,2,3,4,5} with pairwise nonempty intersection.

2023 AIME II #12

Triangles & Polygons▼

Triangle ABC with sides 13,14,15; M midpoint BC; P on circumcircle; find AQ on AM.

2023 AIME II #13

Sequences & Series▼

Acute angle A with tan A = 2 cos A; count n ≤ 1000 where sec^n A + tan^n A ends in 9.

2023 AIME II #14

3D Geometry & Solids▼

Cube with vertices A,B,C,D on plane; heights given; water surface at height 7; find water volume.

2023 AIME II #15

Modular Arithmetic▼

a_n is least positive multiple of 23 congruent to 1 mod 2^n; count n ≤ 1000 with a_n = a_{n+1}.

2022 AIME I #1

Systems & Algebraic Manipulation▼

Find P(0)+Q(0) for two quadratics through (16,54) and (20,53).

2022 AIME I #2

Base Representations▼

Find three-digit integer equal to its base-9 reversal.

2022 AIME I #3

Triangles & Polygons▼

Find distance between intersection points of angle bisectors in isosceles trapezoid.

2022 AIME I #4

Complex Numbers▼

Count ordered pairs (r,s) up to 100 satisfying i*w^r = z^s.

2022 AIME I #5

Systems & Algebraic Manipulation▼

Find distance D between swimmers who meet at north bank simultaneously.

2022 AIME I #6

Counting▼

Count integer pairs (a,b) making sequence strictly increasing with no 4-term AP.

2022 AIME I #7

Factors, Multiples & Divisibility▼

Minimize positive fraction with distinct digits 1-9 in numerator and denominator.

2022 AIME I #8

Circular Geometry▼

Find side length of small equilateral triangle formed by three tangent circles.

2022 AIME I #9

Counting▼

Probability that arrangement of 12 colored blocks has even spacing between same colors.

2022 AIME I #10

3D Geometry & Solids▼

Find AC^2 given three tangent spheres cut by plane into congruent circles.

2022 AIME I #11

Triangles & Polygons▼

Find area of parallelogram given circle tangent to three sides and diagonal lengths.

2022 AIME I #12

Counting▼

Compute ratio S_{2022}/S_{2021} for sum of |A∩B| over equal-size subsets.

2022 AIME I #13

Modular Arithmetic▼

Count distinct numerators of repeating decimals 0.abcd in lowest terms.

2022 AIME I #14

Triangles & Polygons▼

Find perimeter of triangle given splitting lines through midpoints intersect at 30°.

2022 AIME I #15

Systems & Algebraic Manipulation▼

Find [(1-x)(1-y)(1-z)]^2 given three equations with square roots.

2022 AIME II #1

Systems & Algebraic Manipulation▼

Find minimum adults after bus arrives given fractions change.

2022 AIME II #2

Probability & Expected Value▼

Probability Carl wins tennis tournament with given match odds.

2022 AIME II #3

3D Geometry & Solids▼

Find radius of sphere circumscribing a square pyramid.

2022 AIME II #4

Logarithms & Exponents▼

Solve logarithmic equation for value expressed as log base 10.

2022 AIME II #5

Counting▼

Count triangles formed by connecting points with prime differences.

2022 AIME II #6

Inequalities & Optimization▼

Maximize difference between two ordered real numbers with constraints.

2022 AIME II #7

Circular Geometry▼

Find area bounded by three common tangents of two circles.

2022 AIME II #8

Modular Arithmetic▼

Count numbers uniquely determined by floor values of n/4, n/5, n/6.

2022 AIME II #9

Counting▼

Count bounded regions formed by segments connecting points on parallel lines.

2022 AIME II #10

▼

Find remainder of sum of nested binomial coefficients modulo 1000.

2022 AIME II #11

Triangles & Polygons▼

Find square of area of quadrilateral with given sides and angle bisectors.

2022 AIME II #12

Inequalities & Optimization▼

Minimize sum a+b given two ellipse equations with parameters.

2022 AIME II #13

Counting▼

Find coefficient of x^2022 in polynomial defined by rational expression.

2022 AIME II #14

Diophantine Equations▼

Find sum of three least c values with minimum stamp count 97.

2022 AIME II #15

Circular Geometry▼

Find area of hexagon formed by two tangent circles and a third circle.

2021 AIME I #1

Probability & Expected Value▼

Probability Zou wins exactly 5 of 6 races with win/loss dependence

2021 AIME I #2

Triangles & Polygons▼

Area of intersection of two rectangles with given side lengths

2021 AIME I #3

Base Representations▼

Number of positive integers less than 1000 as difference of powers of 2

2021 AIME I #4

Counting▼

Number of ways 66 identical coins split into 3 strictly increasing piles

2021 AIME I #5

Polynomials▼

Sum of third terms of special arithmetic sequences of integers

2021 AIME I #6

3D Geometry & Solids▼

Find AP given distances from P to vertices of a cube

2021 AIME I #7

Trigonometry▼

Number of pairs (m,n) with 1≤m<n≤30 such that sin(mx)+sin(nx)=2

2021 AIME I #8

Functions▼

Number of integers c such that equation has 12 distinct real solutions

2021 AIME I #9

Triangles & Polygons▼

Area of isosceles trapezoid given distances from A to lines

2021 AIME I #10

Modular Arithmetic▼

Sum of indices j where a_j = t/(t+1) in defined sequence

2021 AIME I #11

Triangles & Polygons▼

Perimeter of quadrilateral formed by feet of perpendiculars in cyclic quadrilateral

2021 AIME I #12

Probability & Expected Value▼

Expected minutes until 2 of 3 frogs on dodecagon meet

2021 AIME I #13

Circular Geometry▼

Distance between centers of two circles given third tangent circle conditions

2021 AIME I #14

Modular Arithmetic▼

Least n such that σ(a^n)-1 divisible by 2021 for all positive a

2021 AIME I #15

Systems & Algebraic Manipulation▼

Sum of least and greatest k with 4 intersection points on circle radius ≤21

2021 AIME II #1

Base Representations▼

Find the arithmetic mean of all three-digit palindromes.

2021 AIME II #2

Triangles & Polygons▼

Find AF in an equilateral triangle with given area ratio.

2021 AIME II #3

Counting▼

Count permutations of 1-5 making a sum divisible by 3.

2021 AIME II #4

Complex Numbers▼

Find m+n given shared complex root of two cubics.

2021 AIME II #5

Trigonometry▼

Find a^2+b^2 for s where obtuse triangles are unique.

2021 AIME II #6

Counting▼

Count ordered pairs of subsets satisfying a cardinality equation.

2021 AIME II #7

Systems & Algebraic Manipulation▼

Find m+n from a system of symmetric equations in a,b,c,d.

2021 AIME II #8

Recursion in Combinatorics▼

Probability ant is on top face after 8 moves on cube.

2021 AIME II #9

Modular Arithmetic▼

Count ordered pairs (m,n) with gcd(2^m+1,2^n-1) > 1.

2021 AIME II #10

3D Geometry & Solids▼

Find distance from line to tangency point of three spheres.

2021 AIME II #11

Modular Arithmetic▼

Find sum of greatest elements of possible sets S.

2021 AIME II #12

Trigonometry▼

Find tan of acute angle between diagonals of quadrilateral.

2021 AIME II #13

Modular Arithmetic▼

Find least n such that 2^n+5^n-n is divisible by 1000.

2021 AIME II #14

Triangles & Polygons▼

Find angle BAC given ratios of angles in triangle geometry.

2021 AIME II #15

Functions▼

Find least n such that f(n)/g(n) = 4/7 for recursive functions.

2020 AIME I #1

Triangles & Polygons▼

Isosceles triangle with equal segments, find angle measure sum.

2020 AIME I #2

Logarithms & Exponents▼

Logarithms in geometric progression, find rational x.

2020 AIME I #3

Base Representations▼

Number with base-11 and base-8 representations, find least N.

2020 AIME I #4

Factors, Multiples & Divisibility▼

Numbers ending in 2020 where prefix divides the number.

2020 AIME I #5

Counting▼

Arrangements of 1-6 where removing one yields ascending/descending order.

2020 AIME I #6

3D Geometry & Solids▼

Two spheres in holes of radii 1 and 2, tangent to each other.

2020 AIME I #7

Counting▼

Committee with one more woman than men, count prime factors of N.

2020 AIME I #8

Sequences & Series▼

Bug walks half as far each day, rotates 60°, find OP^2.

2020 AIME I #9

Probability & Expected Value▼

Three divisors of 20^9 chosen with replacement, divisibility probability.

2020 AIME I #10

Modular Arithmetic▼

Positive integers m,n with gcd(m+n,210)=1 and m^m multiple of n^n.

2020 AIME I #11

Polynomials▼

Quadratic f,g with integer coefficients, g(f(2))=g(f(4))=0.

2020 AIME I #12

Factors, Multiples & Divisibility▼

Least n such that 149^n-2^n divisible by 3^3*5^5*7^7.

2020 AIME I #13

Triangles & Polygons▼

Triangle with angle bisector AD, perpendicular bisector meets other bisectors.

2020 AIME I #14

Polynomials▼

Quadratic P with P(P(x))=0 having roots 3,4,a,b, find sum of (a+b)^2.

2020 AIME I #15

Circular Geometry▼

Acute triangle ABC, tangent to circumcircle of HBC meets circumcircle at X,Y.

2020 AIME II #1

Factors, Multiples & Divisibility▼

Find ordered pairs (m,n) of positive integers with m^2 n = 20^20.

2020 AIME II #2

Coordinate Geometry▼

Probability slope from random point to (5/8,3/8) is at least 1/2.

2020 AIME II #3

Logarithms & Exponents▼

Solve log_{2^x} 3^20 = log_{2^{x+3}} 3^2020 for x.

2020 AIME II #4

Triangles & Polygons▼

Find m+x+y for rotation mapping triangle ABC to A'B'C'.

2020 AIME II #5

Base Representations▼

Least n where base-16 digit sum of g(n) uses digit A.

2020 AIME II #6

Sequences & Series▼

Find t_{2020} given recurrence t_n = (5t_{n-1}+1)/(25t_{n-2}).

2020 AIME II #7

3D Geometry & Solids▼

Max r^2 for sphere inside two intersecting right circular cones.

2020 AIME II #8

Functions▼

Least n so sum of zeros of f_n exceeds 500,000.

2020 AIME II #9

Counting▼

Seating orders with no original adjacent pairs sitting together.

2020 AIME II #10

Modular Arithmetic▼

Sum of n where sum of cubes divided by n+5 leaves remainder 17.

2020 AIME II #11

Polynomials▼

Find R(0) given quadratic polynomials with common root conditions.

2020 AIME II #12

Counting▼

Count (m,n) so line through squares 200 and 2000 hits square 1099.

2020 AIME II #13

Triangles & Polygons▼

Area of convex pentagon with incircle and given side lengths.

2020 AIME II #14

Functions▼

Number of solutions to f(f(f(x)))=17 for 0≤x≤2020.

2020 AIME II #15

Triangles & Polygons▼

Find XY^2 given tangents, projections, and sum of squares.

2019 AIME I #1

Sequences & Series▼

Sum of 321 numbers 9, 99, 999, ...; find sum of digits of N.

2019 AIME I #2

Probability & Expected Value▼

Probability that B-J >= 2 for distinct numbers 1-20.

2019 AIME I #3

Triangles & Polygons▼

Area of hexagon formed by points on sides of right triangle.

2019 AIME I #4

Counting▼

Number of ways to make up to 3 substitutions in soccer.

2019 AIME I #5

Probability & Expected Value▼

Probability particle starting at (4,4) first hits axes at origin.

2019 AIME I #6

Triangles & Polygons▼

Find MO in quadrilateral with perpendicular sides and diagonals.

2019 AIME I #7

Logarithms & Exponents▼

Solve system of log equations for x,y; find 3m+2n.

2019 AIME I #8

Polynomials▼

Given sin^10 x + cos^10 x = 11/36, find sin^12 x + cos^12 x.

2019 AIME I #9

Factors, Multiples & Divisibility▼

Sum of six least n with tau(n)+tau(n+1)=7.

2019 AIME I #10

Polynomials▼

Absolute value of sum of pairwise products of distinct roots.

2019 AIME I #11

Triangles & Polygons▼

Minimum perimeter of isosceles triangle with given excircle tangency.

2019 AIME I #12

Complex Numbers▼

Find m+n for z = m+sqrt(n)+11i forming right triangle with f(z).

2019 AIME I #13

Circular Geometry▼

Find BE in triangle with side lengths 4,5,6 and intersecting circles.

2019 AIME I #14

Modular Arithmetic▼

Least odd prime factor of 2019^8 + 1.

2019 AIME I #15

Circular Geometry▼

Find PQ^2 for intersecting circles with chord AB and point P.

2019 AIME II #1

Triangles & Polygons▼

Two congruent triangles overlap; find area of intersection.

2019 AIME II #2

Probability & Expected Value▼

Frog jumps on lily pads; probability it visits pad 7.

2019 AIME II #3

Factors, Multiples & Divisibility▼

Count 7-tuples of positive integers satisfying three product equations.

2019 AIME II #4

Probability & Expected Value▼

Probability product of four dice rolls is a perfect square.

2019 AIME II #5

Counting▼

Seat ambassadors and advisors at a round table with constraints.

2019 AIME II #6

Logarithms & Exponents▼

Solve system of log equations for base b.

2019 AIME II #7

Triangles & Polygons▼

Find perimeter of triangle formed by lines parallel to sides.

2019 AIME II #8

Complex Numbers▼

Evaluate polynomial at complex root; find f(1) mod 1000.

2019 AIME II #9

Factors, Multiples & Divisibility▼

Sum of 20-pretty positive integers less than 2019.

2019 AIME II #10

Trigonometry▼

Find angle theta with alternating sign pattern of tan(2^n theta).

2019 AIME II #11

Circular Geometry▼

Find AK where circles tangent to sides intersect again.

2019 AIME II #12

Recursion in Combinatorics▼

Count progressive sequences with sum 360.

2019 AIME II #13

Coordinate Geometry▼

Find n such that area bounded by PA6, PA7, arc is 1/8 - sqrt2/n.

2019 AIME II #14

Modular Arithmetic▼

Find sum of n where 91 is largest unformable postage.

2019 AIME II #15

Triangles & Polygons▼

Find AB*AC given lengths on orthic triangle chord.

2018 AIME I #1

Counting▼

Count ordered pairs (a,b) so x^2+ax+b factors over integers.

2018 AIME I #2

Base Representations▼

Find base-10 number n given its base-14, base-15, base-6 representations.

2018 AIME I #3

Probability & Expected Value▼

Probability that first 5 cards have all reds and greens adjacent.

2018 AIME I #4

Triangles & Polygons▼

Find AD in isosceles triangle ABC with AD = DE = EC.

2018 AIME I #5

Logarithms & Exponents▼

Find product of all possible K satisfying logarithmic equations.

2018 AIME I #6

Complex Numbers▼

Count complex z with |z|=1 and z^720 - z^120 real.

2018 AIME I #7

Counting▼

Count isosceles triangles from vertices of a right hexagonal prism.

2018 AIME I #8

Triangles & Polygons▼

Find d^2 for largest circle inside equiangular hexagon with given sides.

2018 AIME I #9

Counting▼

Count 4-element subsets of {1..20} with sums 16 and 24.

2018 AIME I #10

Recursion in Combinatorics▼

Count 15-step bug paths on wheel starting and ending at A.

2018 AIME I #11

Modular Arithmetic▼

Least n such that 3^n ends in 01 in base 143.

2018 AIME I #12

Probability & Expected Value▼

Probability sum of random subset of {1..18} is divisible by 3.

2018 AIME I #13

Triangles & Polygons▼

Minimize area of triangle formed by incenters of two subtriangles.

2018 AIME I #14

Recursion in Combinatorics▼

Count sequences of ≤12 jumps on heptagon ending at E.

2018 AIME I #15

Triangles & Polygons▼

Find common area K of three cyclic quadrilaterals with given sine values.

2018 AIME II #1

Systems & Algebraic Manipulation▼

Three runners start at points A, B, C on a line; find AB.

2018 AIME II #2

Sequences & Series▼

Find product of three terms of a recursive sequence modulo 11.

2018 AIME II #3

Base Representations▼

Find sum of bases b where 36_b is square and 27_b is cube.

2018 AIME II #4

Coordinate Geometry▼

Find area enclosed by self-intersecting octagon CORNELIA.

2018 AIME II #5

Complex Numbers▼

Find a^2+b^2 given complex numbers x,y,z with given products.

2018 AIME II #6

Polynomials▼

Probability that polynomial roots are all real for random a.

2018 AIME II #7

Triangles & Polygons▼

Count segments with rational length dividing triangle into equal-area regions.

2018 AIME II #8

Counting▼

Number of jump sequences from (0,0) to (4,4) with step sizes 1 or 2.

2018 AIME II #9

Coordinate Geometry▼

Find area of polygon formed by centroids of 7 triangles in octagon.

2018 AIME II #10

Counting▼

Number of functions f: {1..5}->{1..5} with f(f(x))=f(f(f(x))).

2018 AIME II #11

Counting▼

Number of permutations of 1..6 with a condition on first k terms.

2018 AIME II #12

Triangles & Polygons▼

Find area of convex quadrilateral given side lengths and area condition.

2018 AIME II #13

Probability & Expected Value▼

Probability of rolling an odd number of times to get 1-2-3.

2018 AIME II #14

Triangles & Polygons▼

Find AQ given incircle tangency conditions and side lengths.

2018 AIME II #15

Counting▼

Number of functions f: {0..6}->Z with f(0)=0, f(6)=12, and Lipschitz condition.

2017 AIME I #1

Counting▼

Count triangles with positive area from 15 points on triangle sides.

2017 AIME I #2

Modular Arithmetic▼

Find m+n+r+s given constant remainders for two sets of numbers.

2017 AIME I #3

Modular Arithmetic▼

Find remainder of sum of units digits of triangular numbers.

2017 AIME I #4

3D Geometry & Solids▼

Find volume of pyramid with given base and equal lateral edges.

2017 AIME I #5

Base Representations▼

Find base-ten number abc from base-8 and base-12 representations.

2017 AIME I #6

Probability & Expected Value▼

Find difference between possible vertex angles given chord probability.

2017 AIME I #7

Counting▼

Sum of triple binomial product over nonnegative a,b with a+b≤6.

2017 AIME I #8

Probability & Expected Value▼

Probability that chord QR ≤100 given random angles a,b.

2017 AIME I #9

Modular Arithmetic▼

Find least n>10 such that recursive sequence a_n is multiple of 99.

2017 AIME I #10

Complex Numbers▼

Find real part of z with maximal imaginary part given complex condition.

2017 AIME I #11

Counting▼

Count 3x3 arrangements where median of row medians equals 5.

2017 AIME I #12

Counting▼

Count product-free subsets of {1,...,10}.

2017 AIME I #13

Modular Arithmetic▼

Sum of Q(m) for m=2..2017, where Q(m) is least n with cube property.

2017 AIME I #14

Logarithms & Exponents▼

Find x mod 1000 given logarithmic equations with base a.

2017 AIME I #15

Triangles & Polygons▼

Find minimal area of equilateral triangle inscribed in right triangle.

2017 AIME II #1

Counting▼

Count subsets of {1..8} not subsets of two given 5-element sets.

2017 AIME II #2

Probability & Expected Value▼

Probability that team T4 wins a 4-team playoff with given win probabilities.

2017 AIME II #3

Coordinate Geometry▼

Probability a random point in triangle is closer to vertex B than A or C.

2017 AIME II #4

Base Representations▼

Count positive integers ≤2017 with no digit 0 in base 3.

2017 AIME II #5

Systems & Algebraic Manipulation▼

Maximize sum of two unknown pairwise sums given four known sums.

2017 AIME II #6

Diophantine Equations▼

Sum of positive integers n making sqrt(n^2+85n+2017) an integer.

2017 AIME II #7

Logarithms & Exponents▼

Count integer k in [-500,500] for which log(kx)=2log(x+2) has one real solution.

2017 AIME II #8

Modular Arithmetic▼

Count n<2017 making 1+n+n^2/2!+...+n^6/6! an integer.

2017 AIME II #9

Probability & Expected Value▼

Probability Sharon can discard a card and still have all colors and numbers.

2017 AIME II #10

Coordinate Geometry▼

Area of triangle CDP given rectangle and bisecting segment BP.

2017 AIME II #11

Counting▼

Number of ways to orient roads so every town reachable from every other.

2017 AIME II #12

Triangles & Polygons▼

Distance from center of C0 to point B inside all nested circles.

2017 AIME II #13

Counting▼

Sum of n where f(n+1)=f(n)+78 for isosceles triangle count in n-gon.

2017 AIME II #14

Counting▼

Number of lines containing exactly 8 points in a 10x10x10 grid.

2017 AIME II #15

3D Geometry & Solids▼

Minimum possible sum of distances from a point to tetrahedron vertices.

2016 AIME I #1

Sequences & Series▼

Sum of infinite geometric series S(a) and S(-a) given product 2016.

2016 AIME I #2

Probability & Expected Value▼

Probability of rolling a 7 with weighted dice.

2016 AIME I #3

Counting▼

Number of paths from top to bottom of icosahedron without repeating vertices.

2016 AIME I #4

3D Geometry & Solids▼

Find h^2 given dihedral angle 60° in a right prism pyramid.

2016 AIME I #5

Sequences & Series▼

Find n+t given arithmetic sequences of pages and minutes summing to 374 and 319.

2016 AIME I #6

Triangles & Polygons▼

Find IC in triangle with incenter I and angle bisector meeting circumcircle at D.

2016 AIME I #7

Complex Numbers▼

Number of integer pairs (a,b) making complex number real.

2016 AIME I #8

Counting▼

Minimum sum of three 3-digit numbers from permutation of digits 1-9 with units digit 0.

2016 AIME I #9

Trigonometry▼

Maximum area of rectangle AQRS containing triangle ABC with given sides and sin A.

2016 AIME I #10

Sequences & Series▼

Find a1 given a13=2016 in a sequence alternating geometric and arithmetic.

2016 AIME I #11

Polynomials▼

Find P(7/2) given functional equation (x-1)P(x+1)=(x+2)P(x) and P(2)^2=P(3).

2016 AIME I #12

Factors, Multiples & Divisibility▼

Least positive integer m such that m^2-m+11 is product of at least 4 primes.

2016 AIME I #13

Probability & Expected Value▼

Expected number of jumps for frog to reach river y=24 from (0,21) with fence at y=0.

2016 AIME I #14

Coordinate Geometry▼

Number of squares and circles intersected by line from (0,0) to (1001,429).

2016 AIME I #15

Circular Geometry▼

Find AB^2 given intersecting circles, tangents, and collinear points C, Y, D.

2016 AIME II #1

Systems & Algebraic Manipulation▼

Three people have peanuts in geometric progression; after eating, they form an arithmetic progression.

2016 AIME II #2

Probability & Expected Value▼

Find probability it rains at least one weekend day given conditional probabilities.

2016 AIME II #3

Logarithms & Exponents▼

Solve a system of logarithmic equations for absolute values of logs.

2016 AIME II #4

Factors, Multiples & Divisibility▼

Find minimum volume of a rectangular box with colored unit cubes.

2016 AIME II #5

Triangles & Polygons▼

Find perimeter of right triangle with infinite altitude sum equal to 6 times perimeter.

2016 AIME II #6

Polynomials▼

Find sum of absolute coefficients of product of polynomials.

2016 AIME II #7

Triangles & Polygons▼

Find difference between max and min area of square inscribed in nested squares.

2016 AIME II #8

Counting▼

Count sets of three distinct positive integers with given product.

2016 AIME II #9

Sequences & Series▼

Find middle term of sum of arithmetic and geometric sequences given two terms.

2016 AIME II #10

Circular Geometry▼

Find length ST in cyclic quadrilateral with given segment lengths.

2016 AIME II #11

Factors, Multiples & Divisibility▼

Count numbers less than 1000 that are neither 7-nice nor 8-nice.

2016 AIME II #12

Counting▼

Count colorings of six-section ring with no adjacent same color.

2016 AIME II #13

Counting▼

Find average score of rook placements on 6x6 board.

2016 AIME II #14

3D Geometry & Solids▼

Find distance from point O to vertices of equilateral triangle and two external points.

2016 AIME II #15

Inequalities & Optimization▼

Maximize x_2 given sum and sum of pairwise products constraints.

2015 AIME I #1

Systems & Algebraic Manipulation▼

Find the positive difference between two alternating sum expressions.

2015 AIME I #2

Probability & Expected Value▼

Probability exactly two of three random sleepers are from the same country.

2015 AIME I #3

Diophantine Equations▼

Find prime p such that 16p+1 is a perfect cube.

2015 AIME I #4

Coordinate Geometry▼

Area of triangle formed by midpoints in an equilateral triangle configuration.

2015 AIME I #5

Probability & Expected Value▼

Probability Wednesday is first day Sandy picks matching socks.

2015 AIME I #6

Circular Geometry▼

Find angle measure given arc relationships in two circles.

2015 AIME I #7

Triangles & Polygons▼

Area of a square inscribed in a square with midpoints.

2015 AIME I #8

Base Representations▼

Smallest n with digit sum 20 and n+864 also digit sum 20.

2015 AIME I #9

Counting▼

Count triples (a1,a2,a3) where the generated sequence hits zero.

2015 AIME I #10

Polynomials▼

Find |f(0)| for cubic f with six equal absolute values.

2015 AIME I #11

Triangles & Polygons▼

Smallest perimeter of isosceles triangle with given angle bisector length.

2015 AIME I #12

Counting▼

Arithmetic mean of least elements of all 1000-element subsets of {1..2015}.

2015 AIME I #13

Complex Numbers▼

Product of cosecant squares of odd angles from 1 to 89 degrees.

2015 AIME I #14

Functions▼

Count n where area under floor(sqrt(x)) from 1 to n is integer.

2015 AIME I #15

3D Geometry & Solids▼

Area of unpainted face after slicing a painted cylinder.

2015 AIME II #1

Modular Arithmetic▼

Find remainder when least N satisfying percent conditions is divided by 1000.

2015 AIME II #2

Probability & Expected Value▼

Probability a randomly chosen Latin student is a sophomore.

2015 AIME II #3

Modular Arithmetic▼

Find least positive integer divisible by 17 with digit sum 17.

2015 AIME II #4

Triangles & Polygons▼

Find p+q for perimeter of isosceles trapezoid expressed as log.

2015 AIME II #5

Probability & Expected Value▼

Least n so probability adjacent squares in n×n grid is less than 1/2015.

2015 AIME II #6

Polynomials▼

Sum of two possible c values given polynomial with positive integer roots.

2015 AIME II #7

Triangles & Polygons▼

Find m+n for coefficient beta in area expression of rectangle in triangle.

2015 AIME II #8

Inequalities & Optimization▼

Maximum value of rational expression given inequality on a and b.

2015 AIME II #9

3D Geometry & Solids▼

Volume squared of water displaced by cube with vertical diagonal in cylinder.

2015 AIME II #10

Recursion in Combinatorics▼

Number of quasi-increasing permutations of integers 1 through 7.

2015 AIME II #11

Triangles & Polygons▼

Find m+n for BP given circumcenter and perpendicular line in acute triangle.

2015 AIME II #12

Recursion in Combinatorics▼

Number of 10-letter strings with no more than 3 adjacent identical letters.

2015 AIME II #13

Sequences & Series▼

Index of 100th term where sum of sines from 1 to n is negative.

2015 AIME II #14

Systems & Algebraic Manipulation▼

Evaluate 2x^3 + (xy)^3 + 2y^3 given two symmetric equations.

2015 AIME II #15

Circular Geometry▼

Common area of two triangles formed by line through tangency point of circles.

2014 AIME I #1

Coordinate Geometry▼

Minimum lace length for sneaker with 8 eyelets on a rectangle.

2014 AIME I #2

Probability & Expected Value▼

Find N so probability both balls same color is 0.58.

2014 AIME I #3

Factors, Multiples & Divisibility▼

Number of reduced fractions with numerator and denominator summing to 1000.

2014 AIME I #4

Systems & Algebraic Manipulation▼

Length of trains given passing times for two cyclists.

2014 AIME I #5

Counting▼

Number of communal subsets of a regular 12-gon's vertices.

2014 AIME I #6

Polynomials▼

Find h such that two parabolas have positive integer x-intercepts.

2014 AIME I #7

Complex Numbers▼

Maximum tan^2 of argument of (w-z)/z with |w|=1, |z|=10.

2014 AIME I #8

Modular Arithmetic▼

Four-digit number N such that N and N^2 end in same four digits.

2014 AIME I #9

Polynomials▼

Find x2(x1+x3) for three real roots of a cubic equation.

2014 AIME I #10

Circular Geometry▼

Sin^2 of angle BEA after rolling a small disk around a larger one.

2014 AIME I #11

Probability & Expected Value▼

Probability token ends on |y|=|x| after six random unit moves.

2014 AIME I #12

Counting▼

Probability ranges of two random functions from {1,2,3,4} to itself are disjoint.

2014 AIME I #13

Triangles & Polygons▼

Area of square given perpendicular segments EG=FH=34 and area ratios.

2014 AIME I #14

Systems & Algebraic Manipulation▼

Largest real solution to equation with four rational terms equals quadratic.

2014 AIME I #15

Triangles & Polygons▼

Length DE in triangle ABC with circle through B, D, E, F, G.

2014 AIME II #1

Systems & Algebraic Manipulation▼

Three painters with different rates paint a room; find total minutes.

2014 AIME II #2

Probability & Expected Value▼

Given probabilities of risk factors, find conditional probability as fraction sum.

2014 AIME II #3

Triangles & Polygons▼

Rectangle sides pressed to form hexagon of equal area; find a^2.

2014 AIME II #4

Base Representations▼

Sum of two repeating decimals equals 33/37; find three-digit abc.

2014 AIME II #5

Polynomials▼

Cubic polynomials share roots with shifts; find sum of absolute b values.

2014 AIME II #6

Probability & Expected Value▼

Given two sixes from a fair or biased die, find probability of third six.

2014 AIME II #7

Logarithms & Exponents▼

Sum of logs of function with cosine exponent equals ±1; find sum of n.

2014 AIME II #8

Circular Geometry▼

Three tangent circles with given radii; find radius of circle D as sqrt(m)-n.

2014 AIME II #9

Counting▼

Ten chairs in a circle; count subsets with at least three adjacent chairs.

2014 AIME II #10

Complex Numbers▼

Complex numbers satisfying reciprocal equation; find area of polygon modulo 1000.

2014 AIME II #11

Triangles & Polygons▼

Triangle with given angles and midpoint; find AE as (a-sqrt(b))/c.

2014 AIME II #12

Trigonometry▼

Triangle angles satisfy cos sum condition; find max possible third side length.

2014 AIME II #13

Counting▼

Random pairing of 10 left and right shoes; find probability of no small bad collection.

2014 AIME II #14

Triangles & Polygons▼

Triangle with altitude, angle bisector, median; find AP^2 as reduced fraction.

2014 AIME II #15

Modular Arithmetic▼

Sequence defined by smallest prime not dividing term; find t where x_t=2090.

2013 AIME I #1

Systems & Algebraic Manipulation▼

Tom swims, bikes, and runs a triathlon; find biking time.

2013 AIME I #2

Counting▼

Count five-digit numbers divisible by 5 with equal first/last digits and digit sum divisible by 5.

2013 AIME I #3

Systems & Algebraic Manipulation▼

Square divided by lines into two squares and two rectangles; find sum of ratios.

2013 AIME I #4

Counting▼

Probability a 13-square coloring looks same after 90-degree rotation.

2013 AIME I #5

Polynomials▼

Real root of cubic expressed as sum of cube roots; find sum of constants.

2013 AIME I #6

Probability & Expected Value▼

Probability all 3 math textbooks end up in same box.

2013 AIME I #7

3D Geometry & Solids▼

Rectangular box with given width/length; triangle area of face centers is 30; find height.

2013 AIME I #8

Functions▼

Domain of arcsin(log) is interval of length 1/2013; find minimal m+n mod 1000.

2013 AIME I #9

Triangles & Polygons▼

Equilateral triangle folded so vertex touches side; find fold line length.

2013 AIME I #10

Polynomials▼

Complex root of cubic with integer coefficients; sum of all possible sums of zeros.

2013 AIME I #11

Modular Arithmetic▼

Find sum of distinct prime divisors of least N satisfying block distribution conditions.

2013 AIME I #12

Triangles & Polygons▼

Triangle with hexagon inside; find area of triangle expressed as (a+b√c)/d.

2013 AIME I #13

Triangles & Polygons▼

Triangle with nested similar triangles; find denominator of area of union.

2013 AIME I #14

Sequences & Series▼

Infinite trigonometric series P and Q with given ratio; find sin θ.

2013 AIME I #15

Modular Arithmetic▼

Count ordered triples (A,B,C) satisfying arithmetic sequence and divisibility conditions.

2013 AIME II #1

Systems & Algebraic Manipulation▼

Convert 6:36 AM to new metric time with 10 hours and 100 minutes per day.

2013 AIME II #2

Logarithms & Exponents▼

Find sum of all possible a+b given a logarithmic equation.

2013 AIME II #3

Sequences & Series▼

Find candle height after half the total burn time given burn rates.

2013 AIME II #4

Coordinate Geometry▼

Find product of coordinates of center of equilateral triangle given two vertices.

2013 AIME II #5

Trigonometry▼

Find sine of angle DAE in equilateral triangle with trisected side.

2013 AIME II #6

Modular Arithmetic▼

Find least N such that 1000N to 1000N+999 contains no perfect square.

2013 AIME II #7

Systems & Algebraic Manipulation▼

Find number of files sorted in first 1.5 hours given reassignment pattern.

2013 AIME II #8

Circular Geometry▼

Find radius of circle circumscribing hexagon with given side lengths.

2013 AIME II #9

Counting▼

Count tilings of 7x1 board with colored tiles using all three colors at least once.

2013 AIME II #10

Circular Geometry▼

Find maximum area of triangle BKL given circle and external point A.

2013 AIME II #11

Counting▼

Count functions f from 7-element set to itself such that f(f(x)) is constant.

2013 AIME II #12

Polynomials▼

Count cubic polynomials with integer coefficients whose roots have magnitude 13 or 20.

2013 AIME II #13

Triangles & Polygons▼

Find area of isosceles triangle given lengths from midpoint and trisection point.

2013 AIME II #14

Modular Arithmetic▼

Sum of maximum remainders when n is divided by k for k up to n/2, n=20 to 100.

2013 AIME II #15

Trigonometry▼

Find value of trigonometric expression given two equations involving triangle angles.

2012 AIME I #1

Modular Arithmetic▼

Count three-digit numbers abc where abc and cba are multiples of 4.

2012 AIME I #2

Sequences & Series▼

Find sum of first, last, and middle terms of an arithmetic sequence.

2012 AIME I #3

Counting▼

Count ways waiter serves meals so exactly one person gets correct meal.

2012 AIME I #4

Systems & Algebraic Manipulation▼

Find n+t when Butch and Sundance first meet at a milepost.

2012 AIME I #5

Counting▼

Count subtractions of binary numbers with 5 zeros and 8 ones giving 1.

2012 AIME I #6

Complex Numbers▼

Find n where imaginary part of z is sin(mπ/n) given z^13=w and w^11=z.

2012 AIME I #7

Systems & Algebraic Manipulation▼

Find coins the center student had originally in a network of 16 students.

2012 AIME I #8

3D Geometry & Solids▼

Find p+q where volume of larger solid from plane cut is p/q.

2012 AIME I #9

Logarithms & Exponents▼

Find p+q where xy^5z = 1/2^(p/q) given three equal log expressions.

2012 AIME I #10

Modular Arithmetic▼

Find remainder when 10th smallest element of T is divided by 1000.

2012 AIME I #11

Counting▼

Count reachable points (x,y) with |x|+|y|≤100 via given jumps.

2012 AIME I #12

Triangles & Polygons▼

Find tan B in right triangle with CD and CE trisecting angle C.

2012 AIME I #13

Triangles & Polygons▼

Find largest area of equilateral triangle with vertices on three concentric circles.

2012 AIME I #14

Complex Numbers▼

Find h^2 where h is hypotenuse of right triangle from roots of cubic.

2012 AIME I #15

Modular Arithmetic▼

Count n<1000 where a exists satisfying seating rearrangement conditions.

2012 AIME II #1

Modular Arithmetic▼

Find number of positive integer pairs (m,n) solving 20m+12n=2012.

2012 AIME II #2

Sequences & Series▼

Find a_9 given two geometric sequences with same ratio and a_15=b_11.

2012 AIME II #3

Counting▼

Count committees of 6 with 3 men, 3 women, 2 from each of 3 departments.

2012 AIME II #4

Systems & Algebraic Manipulation▼

Find p+q+r for ratio of field length, width, and distance to point D.

2012 AIME II #5

3D Geometry & Solids▼

Find volume of pyramid formed by folding star figure cut from square.

2012 AIME II #6

Complex Numbers▼

Find c+d for z^4=c+di given |z|=5 and distance maximized.

2012 AIME II #7

Counting▼

Find remainder when 1000th number with 8 binary ones is divided by 1000.

2012 AIME II #8

Complex Numbers▼

Find smallest possible |zw|^2 given system with complex numbers z and w.

2012 AIME II #9

Trigonometry▼

Find p+q for sin2x/sin2y + cos2x/cos2y = p/q given trig ratios.

2012 AIME II #10

Functions▼

Count positive integers n<1000 with positive real x such that n=x floor(x).

2012 AIME II #11

Functions▼

Find m+n for x=m/n satisfying f_{1001}(x)=x-3 with recursive function definition.

2012 AIME II #12

Modular Arithmetic▼

Count numbers ≤10000 that are 7-safe, 11-safe, and 13-safe.

2012 AIME II #13

Trigonometry▼

Find sum of squares of distances from C to four points E_k.

2012 AIME II #14

Counting▼

Count handshake arrangements where each of 9 people shakes exactly 2 hands.

2012 AIME II #15

Triangles & Polygons▼

Find m+n for AF^2=m/n in triangle with inscribed circle and angle bisector.

2011 AIME I #1

Systems & Algebraic Manipulation▼

Mixing acid solutions from three jars to achieve 50% concentration.

2011 AIME I #2

Triangles & Polygons▼

Find length EF inside rectangle with parallel segments.

2011 AIME I #3

Coordinate Geometry▼

Find sum of coordinates of point in rotated coordinate system.

2011 AIME I #4

Triangles & Polygons▼

Find distance between feet of perpendiculars from vertex to angle bisectors.

2011 AIME I #5

Counting▼

Count arrangements of digits 1-9 on nonagon with sum condition.

2011 AIME I #6

Polynomials▼

Find minimum a for parabola with integer value at x=1.

2011 AIME I #7

Modular Arithmetic▼

Count m such that a power equals sum of 2011 other powers.

2011 AIME I #8

Triangles & Polygons▼

Find maximum table height from folded triangle with parallel top.

2011 AIME I #9

Logarithms & Exponents▼

Solve logarithmic equation for cot^2 x in given interval.

2011 AIME I #10

Probability & Expected Value▼

Find sum of n for which obtuse triangle probability is 93/125.

2011 AIME I #11

Modular Arithmetic▼

Sum of all distinct remainders of 2^n modulo 1000.

2011 AIME I #12

Counting▼

Find least women so probability of 4+ consecutive men ≤ 1%.

2011 AIME I #13

3D Geometry & Solids▼

Distance from cube vertex to plane given three adjacent heights.

2011 AIME I #14

Coordinate Geometry▼

Find cos of angle in octagon with perpendicular rays.

2011 AIME I #15

Polynomials▼

Find sum of absolute values of integer roots of cubic.

2011 AIME II #1

Systems & Algebraic Manipulation▼

Fraction of beverage drunk given wasted fraction condition

2011 AIME II #2

Triangles & Polygons▼

Area of square given three equal segments from sides

2011 AIME II #3

Sequences & Series▼

Smallest angle of convex 18-gon with arithmetic sequence angles

2011 AIME II #4

Triangles & Polygons▼

Ratio CP to PA in triangle with angle bisector and midpoint

2011 AIME II #5

Sequences & Series▼

Sum of first 6033 terms of geometric sequence given partial sums

2011 AIME II #6

Counting▼

Number of ordered quadruples with a+d > b+c from 1 to 10

2011 AIME II #7

Counting▼

Maximum red marbles and arrangements with equal neighbor color counts

2011 AIME II #8

Complex Numbers▼

Maximum real part of sum of 12th roots of unity scaled by i

2011 AIME II #9

Inequalities & Optimization▼

Maximum sum of triple products given sum and product constraint

2011 AIME II #10

Circular Geometry▼

Distance from center to intersection of chords in a circle

2011 AIME II #11

Sequences & Series▼

Sum of infinite series involving determinants of tridiagonal matrices

2011 AIME II #12

Probability & Expected Value▼

Probability each delegate sits next to someone from another country

2011 AIME II #13

Triangles & Polygons▼

Length AP on square diagonal given circumcenter angle condition

2011 AIME II #14

Counting▼

Number of permutations of 1..30 with divisibility constraints on positions

2011 AIME II #15

Functions▼

Probability floor of sqrt(P(x)) equals sqrt(P(floor(x))) for random x

2010 AIME I #1

Probability & Expected Value▼

Probability that exactly one of two divisors of 2010^2 is a perfect square.

2010 AIME I #2

Modular Arithmetic▼

Remainder when product of numbers 9, 99, 999, ... is divided by 1000.

2010 AIME I #3

Logarithms & Exponents▼

Find x+y given y = 3x/4 and x^y = y^x.

2010 AIME I #4

Probability & Expected Value▼

Probability Jackie and Phil get same number of heads with biased coin.

2010 AIME I #5

Systems & Algebraic Manipulation▼

Number of possible values of a given a > b > c > d and sum conditions.

2010 AIME I #6

Polynomials▼

Find P(16) for quadratic P(x) bounded by two quadratics with P(11)=181.

2010 AIME I #7

Counting▼

Number of ordered triples of subsets of {1..7} with pairwise intersection 1 and empty triple intersection.

2010 AIME I #8

Coordinate Geometry▼

Minimum radius of disk containing region defined by floor(x)^2 + floor(y)^2 = 25.

2010 AIME I #9

Systems & Algebraic Manipulation▼

Greatest possible value of a^3+b^3+c^3 given system x^3-xyz=2,6,20.

2010 AIME I #10

Base Representations▼

Number of ways to write 2010 as a_3*10^3 + a_2*10^2 + a_1*10 + a_0 with 0 ≤ a_i ≤ 99.

2010 AIME I #11

3D Geometry & Solids▼

Volume of solid formed by revolving region R around line 3y - x = 15.

2010 AIME I #12

Counting▼

Smallest m such that any partition of {3,4,...,m} has a subset with a,b,c where ab=c.

2010 AIME I #13

Triangles & Polygons▼

Find DA given rectangle and semicircle divided by line with area ratio 1:2.

2010 AIME I #14

Functions▼

Largest n such that sum_{k=1}^{100} floor(log10(kn)) ≤ 300.

2010 AIME I #15

Triangles & Polygons▼

Ratio AM/CM in triangle ABC where incircles of ABM and BCM have equal radii.

2010 AIME II #1

Modular Arithmetic▼

Greatest integer multiple of 36 with distinct even digits, remainder mod 1000.

2010 AIME II #2

Probability & Expected Value▼

Probability distance to closest side of unit square is between 1/5 and 1/3.

2010 AIME II #3

Factors, Multiples & Divisibility▼

Greatest n such that 2^n divides product of differences of integers 1 to 20.

2010 AIME II #4

Probability & Expected Value▼

Probability Dave walks at most 400 feet after random gate change.

2010 AIME II #5

Logarithms & Exponents▼

Find sqrt of sum of squares of logs given product and sum of products.

2010 AIME II #6

Polynomials▼

Smallest n so x^4 - nx + 63 factors into nonconstant integer polynomials.

2010 AIME II #7

Complex Numbers▼

Find |a+b+c| for cubic with given complex roots involving w.

2010 AIME II #8

Counting▼

Number of ordered pairs of disjoint nonempty sets covering 1..12 with size constraints.

2010 AIME II #9

Triangles & Polygons▼

Ratio of area of smaller hexagon to larger regular hexagon.

2010 AIME II #10

Polynomials▼

Number of second-degree integer polynomials with integer zeros and f(0)=2010.

2010 AIME II #11

Counting▼

Number of 3x3 matrices with five 1's and four 0's, no two full lines equal.

2010 AIME II #12

Triangles & Polygons▼

Minimum common perimeter of two noncongruent isosceles triangles with same area.

2010 AIME II #13

Probability & Expected Value▼

Minimum probability Alex and Dylan are on same team given they hold a and a+9.

2010 AIME II #14

Triangles & Polygons▼

Ratio AP/BP in right triangle with given angle condition and CP=1.

2010 AIME II #15

Triangles & Polygons▼

Ratio BQ/CQ where Q is intersection of ray AP with BC in triangle ABC.

2009 AIME I #1

Base Representations▼

Find difference between largest and smallest 3-digit numbers with digits in geometric sequence.

2009 AIME I #2

Complex Numbers▼

Find n such that z/(z+n)=4i with Im(z)=164.

2009 AIME I #3

Probability & Expected Value▼

Find p+q given p=m/n and probability of 3 heads equals 1/25 of 5 heads.

2009 AIME I #4

Triangles & Polygons▼

Find AC/AP in parallelogram with points M on AB and N on AD.

2009 AIME I #5

Triangles & Polygons▼

Find LP in triangle with given side lengths and angle bisector.

2009 AIME I #6

Functions▼

Count positive integers N<1000 such that x^{floor(x)}=N has a solution.

2009 AIME I #7

Sequences & Series▼

Find least k>1 such that a_k is integer given recurrence with exponent.

2009 AIME I #8

Sequences & Series▼

Sum of all positive differences of pairs from {2^0,...,2^10} mod 1000.

2009 AIME I #9

Counting▼

Number of guesses for three prizes given digits 1,1,1,1,3,3,3.

2009 AIME I #10

Counting▼

Number of seating arrangements for 15 committee members with restrictions.

2009 AIME I #11

Coordinate Geometry▼

Number of triangles OPQ with integer area, P,Q on line 41x+y=2009.

2009 AIME I #12

Triangles & Polygons▼

Ratio of perimeter of triangle ABI to AB in right triangle with incircle.

2009 AIME I #13

Sequences & Series▼

Minimum possible a1+a2 for sequence defined by recurrence with positive integers.

2009 AIME I #14

Systems & Algebraic Manipulation▼

Minimize S2 given S1=513, S4=4745 for 350 numbers from {1,2,3,4}.

2009 AIME I #15

Triangles & Polygons▼

Maximum area of triangle BPC with incenters and circumcircles.

2009 AIME II #1

Systems & Algebraic Manipulation▼

Find total paint left after painting four equal stripes with given initial amounts.

2009 AIME II #2

Logarithms & Exponents▼

Evaluate sum of three exponential expressions with given logarithmic equations.

2009 AIME II #3

Triangles & Polygons▼

Find greatest integer less than AD in rectangle with perpendicular lines.

2009 AIME II #4

Sequences & Series▼

Find smallest n such that sum of arithmetic sequence equals 2009.

2009 AIME II #5

Circular Geometry▼

Find radius of circle tangent to three circles inscribed in a larger circle.

2009 AIME II #6

Counting▼

Count 5-element subsets of {1..14} with at least two consecutive numbers.

2009 AIME II #7

Factors, Multiples & Divisibility▼

Find ab/10 where denominator of sum of double factorial fractions is 2^a * b.

2009 AIME II #8

Probability & Expected Value▼

Probability that two people rolling for a six have roll counts within one.

2009 AIME II #9

Diophantine Equations▼

Find difference in number of positive integer solutions to two linear Diophantine equations.

2009 AIME II #10

Triangles & Polygons▼

Find distance AD given lighthouses with equal angles and given distances.

2009 AIME II #11

Logarithms & Exponents▼

Find sum of all possible mn given exactly 50 integers k satisfy inequality.

2009 AIME II #12

Counting▼

Maximize number of disjoint pairs from 1..2009 with distinct sums ≤2009.

2009 AIME II #13

Trigonometry▼

Find product of 12 chords in a semicircle divided into 7 equal arcs.

2009 AIME II #14

Sequences & Series▼

Find greatest integer ≤ a_10 given recurrence with square root.

2009 AIME II #15

Circular Geometry▼

Maximize segment length between intersections of chords with diameter in a circle.

2008 AIME I #1

Systems & Algebraic Manipulation▼

Find number of students who like to dance after 20 boys join.

2008 AIME I #2

Triangles & Polygons▼

Find altitude length of isosceles triangle given intersection area with square.

2008 AIME I #3

Systems & Algebraic Manipulation▼

Find sum of squares of biking, jogging, swimming rates.

2008 AIME I #4

Diophantine Equations▼

Find sum of positive integers x and y satisfying quadratic Diophantine equation.

2008 AIME I #5

3D Geometry & Solids▼

Find h/r for cone rolling 17 rotations on its side.

2008 AIME I #6

Factors, Multiples & Divisibility▼

Count entries divisible by 67 in triangular array of odd numbers.

2008 AIME I #7

Factors, Multiples & Divisibility▼

Count sets of 100 consecutive integers containing no perfect square.

2008 AIME I #8

Trigonometry▼

Find n such that sum of arctangents equals pi/4.

2008 AIME I #9

Probability & Expected Value▼

Find probability stack of 10 crates is exactly 41 feet tall.

2008 AIME I #10

Triangles & Polygons▼

Find distance EF in isosceles trapezoid with given diagonals and point E.

2008 AIME I #11

Recursion in Combinatorics▼

Count sequences of A and B of length 14 with run length constraints.

2008 AIME I #12

Systems & Algebraic Manipulation▼

Find maximum number of cars passing a point in one hour given safety rule.

2008 AIME I #13

Polynomials▼

Find common zero of all cubic polynomials vanishing at eight given points.

2008 AIME I #14

Circular Geometry▼

Find maximum possible length of BP in tangent configuration with diameter 18.

2008 AIME I #15

3D Geometry & Solids▼

Find height of paper tray formed by folding cut square corners.

2008 AIME II #1

Sequences & Series▼

Evaluate alternating sum of squares and find remainder mod 1000.

2008 AIME II #2

Systems & Algebraic Manipulation▼

Find total time for two bikers with different speeds and breaks.

2008 AIME II #3

Inequalities & Optimization▼

Maximize remaining volume after ten slices cut from a rectangular block.

2008 AIME II #4

Base Representations▼

Express 2008 as signed sum of powers of 3 and sum exponents.

2008 AIME II #5

Triangles & Polygons▼

Find distance between midpoints of bases in a trapezoid.

2008 AIME II #6

Sequences & Series▼

Find ratio of terms in two recursively defined sequences.

2008 AIME II #7

Polynomials▼

Find sum of cubes of pairwise sums of roots of a cubic.

2008 AIME II #8

Trigonometry▼

Find smallest n making a trigonometric sum an integer.

2008 AIME II #9

Complex Numbers▼

Find position of particle after 150 moves of rotation and translation.

2008 AIME II #10

Counting▼

Count maximum-length paths with strictly increasing distances in a 4x4 grid.

2008 AIME II #11

Triangles & Polygons▼

Find radius of circle tangent to two sides and another circle in a triangle.

2008 AIME II #12

Counting▼

Count arrangements of flags on two poles with no adjacent green flags.

2008 AIME II #13

Triangles & Polygons▼

Find area of image of region outside hexagon under inversion.

2008 AIME II #14

Inequalities & Optimization▼

Maximize a/b given a system of equations with constraints.

2008 AIME II #15

Diophantine Equations▼

Find largest n such that n^2 is difference of consecutive cubes and 2n+79 is square.

2007 AIME I #1

Factors, Multiples & Divisibility▼

Count positive perfect squares less than 10^6 that are multiples of 24.

2007 AIME I #2

Systems & Algebraic Manipulation▼

Find distance from start to middle person when one is halfway between the other two.

2007 AIME I #3

Complex Numbers▼

Find positive real b such that imaginary parts of z^2 and z^3 are equal.

2007 AIME I #4

Modular Arithmetic▼

Find fewest years until three planets and star are collinear again.

2007 AIME I #5

Modular Arithmetic▼

Count integer Fahrenheit temperatures where rounding conversion yields original value.

2007 AIME I #6

Recursion in Combinatorics▼

Count move sequences from 0 to 39 using multiples of 3 or 13.

2007 AIME I #7

Sequences & Series▼

Find remainder when sum involving floor/ceiling of log base sqrt(2) is divided by 1000.

2007 AIME I #8

Polynomials▼

Find largest k such that two quadratics are both factors of a cubic.

2007 AIME I #9

Triangles & Polygons▼

Find radius of two equal circles tangent to hypotenuse and leg extensions.

2007 AIME I #10

Counting▼

Count shadings of 6x4 grid with 2 shaded per row and 3 per column.

2007 AIME I #11

Modular Arithmetic▼

Sum over p of integer nearest sqrt(p) for p=1 to 2007, mod 1000.

2007 AIME I #12

Trigonometry▼

Find area common to original and rotated isosceles triangle, then compute expression.

2007 AIME I #13

3D Geometry & Solids▼

Find area of plane intersection with square pyramid, expressed as sqrt(p).

2007 AIME I #14

Sequences & Series▼

Find greatest integer ≤ (a_2006^2 + a_2007^2)/(a_2006 a_2007) for given recurrence.

2007 AIME I #15

Triangles & Polygons▼

Find r in side length expression p ± q√r for equilateral triangle with interior 60° angle.

2007 AIME II #1

Counting▼

Count license plates with characters from AIME and 2007.

2007 AIME II #2

Factors, Multiples & Divisibility▼

Count ordered triples (a,b,c) with a|b, a|c, a+b+c=100.

2007 AIME II #3

Triangles & Polygons▼

Find EF^2 in a square with external right triangles.

2007 AIME II #4

Systems & Algebraic Manipulation▼

Find m given production rates of widgets and whoosits.

2007 AIME II #5

Modular Arithmetic▼

Count unit squares below line 9x+223y=2007 in first quadrant.

2007 AIME II #6

Counting▼

Count four-digit parity-monotonic integers.

2007 AIME II #7

Factors, Multiples & Divisibility▼

Find max n_i/k given floor cube root and divisibility conditions.

2007 AIME II #8

Inequalities & Optimization▼

Maximize basic rectangles from lines on a 4x5 paper.

2007 AIME II #9

Triangles & Polygons▼

Find PQ where incircles of two triangles touch EF.

2007 AIME II #10

Probability & Expected Value▼

Probability that B is subset of A or complement of A.

2007 AIME II #11

Circular Geometry▼

Distance large tube rolls over small tube without slipping.

2007 AIME II #12

Sequences & Series▼

Find log_3(x_14) given sum of logs and sum bounds.

2007 AIME II #13

Counting▼

Count bottom row 0/1 distributions making top sum multiple of 3.

2007 AIME II #14

Polynomials▼

Find f(5) given functional equation and values.

2007 AIME II #15

Circular Geometry▼

Find radius of circle tangent to three equal circles in triangle.

2006 AIME I #1

Triangles & Polygons▼

Find the perimeter of quadrilateral ABCD with right angles and perpendicular diagonals.

2006 AIME I #2

Counting▼

Find the number of possible sums of a 90-element subset of 1 to 100.

2006 AIME I #3

Base Representations▼

Find the least positive integer such that deleting its leftmost digit yields 1/29 of it.

2006 AIME I #4

Factors, Multiples & Divisibility▼

Find the remainder when the number of trailing zeros of 1!2!...100! is divided by 1000.

2006 AIME I #5

Systems & Algebraic Manipulation▼

Find abc where sqrt(104√6+468√10+144√15+2006) = a√2+b√3+c√5.

2006 AIME I #6

Counting▼

Find the sum of all numbers 0.abc with distinct digits a,b,c.

2006 AIME I #7

Coordinate Geometry▼

Find ratio of area D to area A given ratio of C to B is 11/5.

2006 AIME I #8

Triangles & Polygons▼

Find number of possible integer areas K of rhombus T given four congruent rhombi.

2006 AIME I #9

Logarithms & Exponents▼

Find number of ordered pairs (a,r) given log_8 sum of geometric sequence equals 2006.

2006 AIME I #10

Coordinate Geometry▼

Find a^2+b^2+c^2 for line with slope 3 dividing eight circles into equal area.

2006 AIME I #11

Recursion in Combinatorics▼

Find remainder when number of towers from 8 cubes with edge-length rule is divided by 1000.

2006 AIME I #12

Trigonometry▼

Find sum of x in (100,200) satisfying cos^3 3x+cos^3 5x=8 cos^3 4x cos^3 x.

2006 AIME I #13

Factors, Multiples & Divisibility▼

Find greatest n<1000 such that sum of greatest powers of 2 dividing even numbers is a perfect square.

2006 AIME I #14

3D Geometry & Solids▼

Find floor(m+√n) for height of broken tripod with legs of length 5.

2006 AIME I #15

Sequences & Series▼

Find minimum possible absolute sum of sequence with |x_k|=|x_{k-1}+3| and x_0=0.

2006 AIME II #1

Triangles & Polygons▼

Find side length of convex hexagon with congruent sides and right angles.

2006 AIME II #2

Logarithms & Exponents▼

Count integer n such that logs of 12, 75, n form triangle.

2006 AIME II #3

Factors, Multiples & Divisibility▼

Find largest k such that product of first 100 odd numbers divisible by 3^k.

2006 AIME II #4

Counting▼

Count permutations of 1..12 with decreasing then increasing sequence.

2006 AIME II #5

Probability & Expected Value▼

Find probability of face F on unfair die given sum 7 probability.

2006 AIME II #6

Triangles & Polygons▼

Find side length of square inside unit square with equilateral triangle.

2006 AIME II #7

Counting▼

Count ordered pairs (a,b) with a+b=1000 and neither has zero digit.

2006 AIME II #8

Counting▼

Count distinguishable colorings of large triangle made of four small triangles.

2006 AIME II #9

Coordinate Geometry▼

Find x-coordinate of intersection of two common internal tangents to circles.

2006 AIME II #10

Probability & Expected Value▼

Probability team A finishes with more points than B after first game win.

2006 AIME II #11

Sequences & Series▼

Find remainder of sum of first 28 terms of tribonacci-like sequence mod 1000.

2006 AIME II #12

Triangles & Polygons▼

Find area of triangle CBG in circle with equilateral triangle and parallels.

2006 AIME II #13

Factors, Multiples & Divisibility▼

Count integers less than 1000 expressible as sum of j consecutive odd integers in exactly 5 ways.

2006 AIME II #14

Base Representations▼

Find smallest n such that sum of reciprocals of non-zero digits from 1 to 10^n is integer.

2006 AIME II #15

Systems & Algebraic Manipulation▼

Find x+y+z given three equations involving square roots of squares minus constants.

2005 AIME I #1

Circular Geometry▼

Six congruent circles in a ring inside a larger circle; find floor of area between them.

2005 AIME I #2

Factors, Multiples & Divisibility▼

Count values of k such that arithmetic sequence S_k contains 2005.

2005 AIME I #3

Factors, Multiples & Divisibility▼

Count positive integers with exactly three proper divisors each less than 50.

2005 AIME I #4

Diophantine Equations▼

Find maximum band members given square formation leaves 5, rectangular formation has 7 more rows than columns.

2005 AIME I #5

Counting▼

Count distinguishable stacks of 4 gold and 4 silver coins with no adjacent faces touching.

2005 AIME I #6

Polynomials▼

Find floor of product of nonreal roots of x^4-4x^3+6x^2-4x=2005.

2005 AIME I #7

Triangles & Polygons▼

Find AB in quadrilateral with given sides and two 60-degree angles.

2005 AIME I #8

Polynomials▼

Find sum of three real roots of exponential equation in terms of m/n.

2005 AIME I #9

Probability & Expected Value▼

Probability that entire surface of 3x3x3 cube is orange after random arrangement.

2005 AIME I #10

Coordinate Geometry▼

Find largest possible p+q for vertex A of triangle with given area and median slope.

2005 AIME I #11

Circular Geometry▼

Find maximum diameter of semicircle inside an 8x8 square.

2005 AIME I #12

Factors, Multiples & Divisibility▼

Find |a-b| where a counts n with odd S(n) and b counts n with even S(n) up to 2005.

2005 AIME I #13

Counting▼

Count paths from (0,0) to (5,5) with moves right, up, or diagonal and no right-angle turns.

2005 AIME I #14

Coordinate Geometry▼

Find area of square with each of four given points on a different side.

2005 AIME I #15

Triangles & Polygons▼

Find area of triangle with BC=20 and incircle trisecting median AD.

2005 AIME II #1

Counting▼

Find n such that C(n,6) = 6 * C(n,3).

2005 AIME II #2

Probability & Expected Value▼

Probability each of three guests gets one of each roll type.

2005 AIME II #3

Sequences & Series▼

Infinite geometric series with sum 2005; squaring each term gives series with 10x the sum. Common ratio m/n (relatively prime); find m+n.

2005 AIME II #4

Factors, Multiples & Divisibility▼

Count divisors of at least one of 10^10, 15^7, 18^11.

2005 AIME II #5

Logarithms & Exponents▼

Count ordered pairs (a,b) satisfying log_a b + 6 log_b a = 5.

2005 AIME II #6

Counting▼

Find number of cards where card 131 retains original position.

2005 AIME II #7

Systems & Algebraic Manipulation▼

Simplify expression involving nested radicals of 5, find (x+1)^48.

2005 AIME II #8

Circular Geometry▼

Find chord length of circle C3 tangent to C1 and C2.

2005 AIME II #9

Complex Numbers▼

Count n ≤ 1000 where (sin t + i cos t)^n = sin nt + i cos nt.

2005 AIME II #10

3D Geometry & Solids▼

Ratio of volume of regular octahedron to its dual cube.

2005 AIME II #11

Sequences & Series▼

Find m given recurrence a_{k+1} = a_{k-1} - 3/a_k with a_0=37, a_1=72, a_m=0.

2005 AIME II #12

Triangles & Polygons▼

Find BF in square ABCD with given lengths and 45° angle.

2005 AIME II #13

Polynomials▼

Find product of integer solutions n to P(n)=n+3 given P(17)=10, P(24)=17.

2005 AIME II #14

Triangles & Polygons▼

Find BE in triangle ABC with given side lengths and equal angles.

2005 AIME II #15

Coordinate Geometry▼

Find smallest positive a so line y=ax contains center of circle tangent to two given circles.

2004 AIME I #1

Modular Arithmetic▼

Sum of remainders when four-digit numbers with consecutive decreasing digits are divided by 37.

2004 AIME I #2

Systems & Algebraic Manipulation▼

Find m given sums of consecutive integers in two sets and difference of largest elements.

2004 AIME I #3

Counting▼

Count space diagonals of a convex polyhedron given vertices, edges, and faces.

2004 AIME I #4

Coordinate Geometry▼

Area enclosed by midpoints of length-2 segments on adjacent sides of a square.

2004 AIME I #5

Inequalities & Optimization▼

Largest possible two-day success ratio for Beta given constraints on daily scores.

2004 AIME I #6

Counting▼

Count four-digit snakelike numbers with distinct digits between 1000 and 9999.

2004 AIME I #7

Polynomials▼

Find absolute value of coefficient of x^2 in product (1-x)(1+2x)...(1-15x).

2004 AIME I #8

Modular Arithmetic▼

Number of non-similar regular 1000-pointed stars.

2004 AIME I #9

Triangles & Polygons▼

Minimum area of triangle similar to 3-4-5 triangle cut from rectangle.

2004 AIME I #10

Coordinate Geometry▼

Probability that a unit circle placed in a 15-by-36 rectangle does not touch diagonal.

2004 AIME I #11

3D Geometry & Solids▼

Find k where painted surface area ratio equals volume ratio for cone cut by parallel plane.

2004 AIME I #12

Functions▼

Area of set where floor(log2(1/x)) and floor(log5(1/y)) are both even.

2004 AIME I #13

Complex Numbers▼

Sum of five smallest arguments of complex roots of polynomial P(x).

2004 AIME I #14

3D Geometry & Solids▼

Length of rope touching cylindrical tower when unicorn pulls it taut.

2004 AIME I #15

Recursion in Combinatorics▼

Number of positive integers x such that d(x)=20 under given function f.

2004 AIME II #1

Circular Geometry▼

Ratio of areas formed by a chord perpendicular to a radius at its midpoint.

2004 AIME II #2

Probability & Expected Value▼

Probability Terry and Mary pick same color combination of candies.

2004 AIME II #3

Factors, Multiples & Divisibility▼

Minimize number of 1-cm cubes in a rectangular block with 231 invisible cubes.

2004 AIME II #4

Counting▼

Count positive integers less than 10000 with at most two different digits.

2004 AIME II #5

Systems & Algebraic Manipulation▼

Minimum additional workers needed to complete a job on schedule after layoffs.

2004 AIME II #6

Systems & Algebraic Manipulation▼

Least total bananas given ratios after three monkeys take and share.

2004 AIME II #7

Triangles & Polygons▼

Perimeter of rectangle after folding corner B to point B' on AD.

2004 AIME II #8

Factors, Multiples & Divisibility▼

Number of divisors of 2004^2004 divisible by exactly 2004 positive integers.

2004 AIME II #9

Sequences & Series▼

Find n + a_n for greatest term less than 1000 in a mixed progression sequence.

2004 AIME II #10

Modular Arithmetic▼

Probability a number with two 1's in binary up to 2^40 is divisible by 9.

2004 AIME II #11

3D Geometry & Solids▼

Shortest distance a fly crawls on a cone surface between two points.

2004 AIME II #12

Circular Geometry▼

Radius of circle tangent to four circles centered at trapezoid vertices.

2004 AIME II #13

Triangles & Polygons▼

Ratio of areas of triangles ABC and EBD in a convex pentagon with parallels.

2004 AIME II #14

Diophantine Equations▼

Number of n such that inserting + into n 7's yields 7000.

2004 AIME II #15

Recursion in Combinatorics▼

Squares below the original 942nd square after 10 folds of a 1024 strip.

2003 AIME I #1

Systems & Algebraic Manipulation▼

Simplify factorial expression to find largest possible n and k+n.

2003 AIME I #2

Sequences & Series▼

Find ratio of green area to largest circle in concentric circles.

2003 AIME I #3

Counting▼

Sum of larger element in each two-element subset of a set.

2003 AIME I #4

Logarithms & Exponents▼

Solve logarithmic equations involving sin and cos to find n.

2003 AIME I #5

3D Geometry & Solids▼

Volume of points within one unit of a 3x4x5 box.

2003 AIME I #6

3D Geometry & Solids▼

Sum of areas of all triangles with vertices of a unit cube.

2003 AIME I #7

Diophantine Equations▼

Find sum of perimeters of isosceles triangle with integer sides.

2003 AIME I #8

Sequences & Series▼

Find sum of four terms in arithmetic and geometric progression.

2003 AIME I #9

Counting▼

Count 4-digit numbers where sum of first two digits equals last two.

2003 AIME I #10

Triangles & Polygons▼

Find angle CMB in isosceles triangle with given interior angles.

2003 AIME I #11

Trigonometry▼

Probability sin^2 x, cos^2 x, sin x cos x do not form a triangle.

2003 AIME I #12

Triangles & Polygons▼

Find floor of 1000 cos A in quadrilateral with equal angles and sides.

2003 AIME I #13

Counting▼

Count numbers ≤2003 with more 1s than 0s in binary, mod 1000.

2003 AIME I #14

Base Representations▼

Find smallest denominator n so decimal of m/n contains 251 consecutively.

2003 AIME I #15

Triangles & Polygons▼

Find DE:EF ratio in triangle with angle bisector and median.

2003 AIME II #1

Systems & Algebraic Manipulation▼

Find sum of all possible products of three positive integers with given conditions.

2003 AIME II #2

Modular Arithmetic▼

Find remainder when greatest integer multiple of 8 with distinct digits is divided by 1000.

2003 AIME II #3

Counting▼

Count seven-letter words from A, B, C with no forbidden adjacent pairs.

2003 AIME II #4

3D Geometry & Solids▼

Find ratio of volumes of tetrahedron formed by face centers to original tetrahedron.

2003 AIME II #5

3D Geometry & Solids▼

Find volume of wedge cut from a cylindrical log by two planar cuts.

2003 AIME II #6

Triangles & Polygons▼

Find area of union of triangle and its 180-degree rotation about centroid.

2003 AIME II #7

Triangles & Polygons▼

Find area of rhombus given circumradii of two triangles formed by its diagonals.

2003 AIME II #8

Sequences & Series▼

Find eighth term of sequence formed by multiplying corresponding terms of two arithmetic sequences.

2003 AIME II #9

Polynomials▼

Find sum of P evaluated at roots of Q, where P and Q are given polynomials.

2003 AIME II #10

Diophantine Equations▼

Find maximum possible sum of two positive integers differing by 60 with given square root condition.

2003 AIME II #11

Triangles & Polygons▼

Find area of triangle CDM in right triangle ABC with given side lengths and point D.

2003 AIME II #12

Inequalities & Optimization▼

Find smallest possible number of committee members given vote percentage condition.

2003 AIME II #13

Recursion in Combinatorics▼

Find probability bug returns to starting vertex after ten moves on equilateral triangle.

2003 AIME II #14

Coordinate Geometry▼

Find area of convex equilateral hexagon with given angle and parallel side conditions.

2003 AIME II #15

Polynomials▼

Find sum of absolute imaginary parts of squares of distinct zeros of given polynomial.

2002 AIME I #1

Probability & Expected Value▼

Probability a license plate has a three-letter or three-digit palindrome.

2002 AIME I #2

Circular Geometry▼

Ratio of rectangle dimensions given tangent circles.

2002 AIME I #3

Base Representations▼

Number of ordered pairs (Dick's age, years) with digit reversal.

2002 AIME I #4

Sequences & Series▼

Find m+n given telescoping sum equals 1/29.

2002 AIME I #5

Counting▼

Number of squares with at least two vertices of a regular dodecagon.

2002 AIME I #6

Logarithms & Exponents▼

Evaluate log base 30 of product of solutions to log system.

2002 AIME I #7

Modular Arithmetic▼

First three decimal digits of (10^2002+1)^(10/7).

2002 AIME I #8

Sequences & Series▼

Smallest k such that more than one Fibonacci-like sequence exists.

2002 AIME I #9

Modular Arithmetic▼

Sum of paintable integers 100h+10t+u for fence painting.

2002 AIME I #10

Triangles & Polygons▼

Area of quadrilateral DCFG in right triangle with angle bisector.

2002 AIME I #11

3D Geometry & Solids▼

Length of light path reflecting off cube faces until vertex.

2002 AIME I #12

Complex Numbers▼

Find a+b for complex iteration z_n = F(z_{n-1}) with period 3.

2002 AIME I #13

Triangles & Polygons▼

Area of triangle AFB with medians and circumcircle.

2002 AIME I #14

Modular Arithmetic▼

Maximum size of set S with integer mean after deleting any element.

2002 AIME I #15

3D Geometry & Solids▼

Compute EG^2 for polyhedron with square, trapezoid, and triangle faces.

2002 AIME II #1

Base Representations▼

Count distinct absolute differences between three-digit numbers and their reversals.

2002 AIME II #2

3D Geometry & Solids▼

Find surface area of a cube given three vertices.

2002 AIME II #3

Logarithms & Exponents▼

Find sum of three positive integers in geometric sequence with given log condition.

2002 AIME II #4

Sequences & Series▼

Find area of garden enclosed by hexagonal blocks with n=202.

2002 AIME II #5

Factors, Multiples & Divisibility▼

Sum positive integers of form 2^n 3^m where expression is not a divisor.

2002 AIME II #6

Sequences & Series▼

Find integer closest to 1000 times sum of 1/(n^2-4) from n=3 to 10000.

2002 AIME II #7

Modular Arithmetic▼

Find smallest positive k such that sum of squares is multiple of 200.

2002 AIME II #8

Modular Arithmetic▼

Find least positive integer k with no integer n satisfying floor(2002/n)=k.

2002 AIME II #9

Counting▼

Count unordered pairs of non-empty disjoint subsets of {1,...,10}.

2002 AIME II #10

Trigonometry▼

Find sum m+n+p+q for two least positive x with sin(x deg)=sin(x rad).

2002 AIME II #11

Sequences & Series▼

Find second term of two geometric series with sum 1 and same second term.

2002 AIME II #12

Probability & Expected Value▼

Probability shooting average is 0.4 after 10 shots and never exceeds 0.4.

2002 AIME II #13

Triangles & Polygons▼

Find ratio of area of triangle PQR to area of triangle ABC.

2002 AIME II #14

Triangles & Polygons▼

Find OP given triangle perimeter, right angle, and inscribed circle tangent to sides.

2002 AIME II #15

Coordinate Geometry▼

Find m in y=mx tangent to two circles intersecting at (9,6) with radii product 68.

2001 AIME I #1

Factors, Multiples & Divisibility▼

Sum of two-digit integers divisible by each of their digits

2001 AIME I #2

Systems & Algebraic Manipulation▼

Mean of a set given means after adding 1 and 2001

2001 AIME I #3

Polynomials▼

Sum of all roots of x^2001 + (1/2 - x)^2001 = 0

2001 AIME I #4

Triangles & Polygons▼

Area of triangle with given angles and angle bisector length

2001 AIME I #5

Coordinate Geometry▼

Side length of equilateral triangle inscribed in ellipse

2001 AIME I #6

Probability & Expected Value▼

Probability that four die rolls are non-decreasing

2001 AIME I #7

Triangles & Polygons▼

Length of segment through incenter parallel to base

2001 AIME I #8

Base Representations▼

Largest integer whose base-7 digits equal twice the number in base-10

2001 AIME I #9

Triangles & Polygons▼

Ratio of areas of triangle DEF to ABC given side ratios

2001 AIME I #10

Probability & Expected Value▼

Probability that midpoint of two lattice points is also a lattice point

2001 AIME I #11

Modular Arithmetic▼

Smallest N for consistent renumbering of a 5xN array

2001 AIME I #12

3D Geometry & Solids▼

Inradius of tetrahedron with vertices at (6,0,0), (0,4,0), (0,0,2), (0,0,0)

2001 AIME I #13

Trigonometry▼

Chord length of 3d-degree arc given chord lengths of d and 2d arcs

2001 AIME I #14

Recursion in Combinatorics▼

Number of 19-house mail patterns with no two adjacent and no three empty

2001 AIME I #15

Counting▼

Probability no consecutive numbers on adjacent octahedron faces

2001 AIME II #1

Base Representations▼

Largest integer where each consecutive digit pair forms a perfect square.

2001 AIME II #2

Counting▼

Find difference between max and min students studying both languages.

2001 AIME II #3

Sequences & Series▼

Sum of three terms of a recursively defined sequence.

2001 AIME II #4

Coordinate Geometry▼

Length of segment PQ with R as midpoint on given lines.

2001 AIME II #5

Counting▼

Largest n so all ten-element subsets of {4,...,n} have triangle property.

2001 AIME II #6

Circular Geometry▼

Area of square inscribed in circle with another square on its side.

2001 AIME II #7

Triangles & Polygons▼

Distance between centers of incircles of two triangles in a right triangle.

2001 AIME II #8

Functions▼

Smallest x such that f(x)=f(2001) given functional equation and piecewise definition.

2001 AIME II #9

Probability & Expected Value▼

Probability a 3x3 grid has no 2x2 red square when each cell is colored randomly.

2001 AIME II #10

Modular Arithmetic▼

Number of positive multiples of 1001 of the form 10^j - 10^i.

2001 AIME II #11

Probability & Expected Value▼

Probability soccer team has more wins than losses in 6 games with equal win/loss/tie odds.

2001 AIME II #12

3D Geometry & Solids▼

Volume after recursively adding tetrahedra to faces of a regular tetrahedron.

2001 AIME II #13

Triangles & Polygons▼

Length CD in quadrilateral with given angle equalities and side lengths.

2001 AIME II #14

Complex Numbers▼

Sum of even-indexed arguments of complex numbers on unit circle satisfying z^28 - z^8 - 1 = 0.

2001 AIME II #15

3D Geometry & Solids▼

Surface area of cube with triangular tunnel drilled through it.

2000 AIME I #1

Factors, Multiples & Divisibility▼

Least n such that any factorization of 10^n has a factor with digit 0.

2000 AIME I #2

Coordinate Geometry▼

Area of pentagon formed by reflections of (u,v) is 451; find u+v.

2000 AIME I #3

Polynomials▼

Coefficients of x^2 and x^3 in (ax+b)^2000 equal; find a+b.

2000 AIME I #4

Systems & Algebraic Manipulation▼

Rectangle dissected into nine squares; find perimeter given relatively prime sides.

2000 AIME I #5

Probability & Expected Value▼

Probability both marbles white given total 25 and black probability 27/50.

2000 AIME I #6

Systems & Algebraic Manipulation▼

Number of integer pairs (x,y) with 0<x<y<10^6 and arithmetic mean 2 more than geometric mean.

2000 AIME I #7

Systems & Algebraic Manipulation▼

Given xyz=1, x+1/z=5, y+1/x=29, find z+1/y as reduced fraction.

2000 AIME I #8

3D Geometry & Solids▼

Height of liquid in inverted cone given original depth 9 inches.

2000 AIME I #9

Logarithms & Exponents▼

System of log equations with two solutions; find sum of y-values.

2000 AIME I #10

Sequences & Series▼

Sequence where x_k is k less than sum of other 99 terms; find x_50.

2000 AIME I #11

Factors, Multiples & Divisibility▼

Sum of all a/b with a,b relatively prime divisors of 1000; floor of S/10.

2000 AIME I #12

Functions▼

Function symmetric about three lines; max distinct values from 0 to 999.

2000 AIME I #13

Coordinate Geometry▼

Area reachable by firetruck in 6 minutes on highways and prairie.

2000 AIME I #14

Triangles & Polygons▼

Isosceles triangle with AP=PQ=QB=BC; find floor(1000 * angle ratio).

2000 AIME I #15

Recursion in Combinatorics▼

Card shuffling process yields ascending order; find cards above 1999 originally.

2000 AIME II #1

Logarithms & Exponents▼

Simplify sum of fractions with logarithms to find m+n.

2000 AIME II #2

Diophantine Equations▼

Count lattice points on hyperbola x^2 - y^2 = 2000^2.

2000 AIME II #3

Probability & Expected Value▼

Probability two cards form a pair after one pair removed.

2000 AIME II #4

Factors, Multiples & Divisibility▼

Find smallest integer with 6 odd and 12 even divisors.

2000 AIME II #5

Counting▼

Count arrangements of rings on fingers with order significance.

2000 AIME II #6

Triangles & Polygons▼

Find x^2/100 floor for trapezoid with equal area division.

2000 AIME II #7

Counting▼

Evaluate sum of reciprocals of factorials to find N/100 floor.

2000 AIME II #8

Triangles & Polygons▼

Find BC^2 in trapezoid with perpendicular diagonals.

2000 AIME II #9

Complex Numbers▼

Find least integer greater than |z| for complex z satisfying equation.

2000 AIME II #10

Circular Geometry▼

Find square of radius of circle inscribed in quadrilateral.

2000 AIME II #11

Coordinate Geometry▼

Sum absolute slopes of AB in isosceles trapezoid with integer vertices.

2000 AIME II #12

3D Geometry & Solids▼

Find distance from sphere center to triangle ABC on its surface.

2000 AIME II #13

Polynomials▼

Find m+n+r for real root of polynomial expressed as (m+√n)/r.

2000 AIME II #14

Base Representations▼

Evaluate alternating sum of factorial base digits of large expression.

2000 AIME II #15

Trigonometry▼

Find least n such that sum of cosecant products equals 1/sin n°.

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