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AIME I 2026

Problems, Step-by-Step Hints, and Solutions

1.
Patrick started walking at a constant speed along a straight road from his school to the park. One hour after Patrick left, Tanya started running at a constant speed of 2 miles per hour faster than Patrick walked, following the same straight road from the school to the park. One hour after Tanya left, Jose started bicycling at a constant speed of 7 miles per hour faster than Tanya ran, following the same straight road from the school to the park. All three people arrived at the park at the same time. The distance from the school to the park is mn\frac{m}{n} miles, where mm and nn are relatively prime positive integers. Find m+nm+n.
2.
Find the number of positive integer palindromes written in base 10, with no zero digits, and whose digits add up to 13. For example, 42124 has these properties. Recall that a palindrome is a number whose representation reads the same from left to right as from right to left.
3.
A hemisphere with radius 200 sits on top of a horizontal circular disk with radius 200, and the hemisphere and disk have the same center. Let TT be the region of points PP in the disk such that a sphere of radius 42 can be placed on top of the disk at PP and lie completely inside the hemisphere. The area of TT divided by the area of the disk is pq\frac{p}{q}, where pp and qq are relatively prime positive integers. Find p+qp+q.
4.
Find the number of integers less than or equal to 100100 that are equal to a+b+aba+b+ab for some choice of distinct positive integers aa and bb.
5.
A plane contains points AA and BB with AB=1AB=1. Point AA is rotated in the plane counterclockwise through an acute angle θ\theta around point BB to point A′A'. Then BB is rotated in the plane clockwise through angle θ\theta around point A′A' to point B′B'. Suppose AB′=43A B'=\frac43. The value of cos⁡θ\cos\theta can be written as mn\frac mn, where mm and nn are relatively prime positive integers. Find m+nm+n.
6.
The product of all positive real numbers xx satisfying the equationxlog⁡2026x20=26x\sqrt[20]{x^{\log_{2026}x}}=26xis an integer PP. Find the number of positive integer divisors of PP.
7.
Find the number of functions π\pi mapping the set A={1,2,3,4,5,6}A=\{1,2,3,4,5,6\} onto AA such that for every a∈Aa\in A,
π(π(π(π(π(π(a))))))=a.\pi(\pi(\pi(\pi(\pi(\pi(a))))))=a.
8.
Let NN be the number of positive integer divisors of 170171717017^{17} that leave a remainder of 55 upon division by 1212. Find the remainder when NN is divided by 10001000.
9.
Joanne has a blank fair six-sided die and six stickers each displaying a different integer from 1 to 6. Joanne rolls the die and then places the sticker labeled 1 on the top face of the die. She then rolls the die again, places the sticker labeled 2 on the top face, and continues this process to place the rest of the stickers in order. If the die ever lands with a sticker already on its top face, the new sticker is placed to cover the old sticker.

Let pp be the conditional probability that at the end of the process exactly one face has been left blank, given that all the even-numbered stickers are visible on faces of the die. Then pp can be written as mn\frac{m}{n}, where mm and nn are relatively prime positive integers. Find m+nm+n.
10.
Let △ABC\triangle ABC have side lengths AB=13AB=13, BC=14BC=14, and CA=15CA=15. Triangle A′B′C′A'B'C' is obtained by rotating △ABC\triangle ABC about its circumcenter so that A′C′A'C' is perpendicular to BCBC, with A′A' and BB not on the same side of line B′C′B'C'. Find the integer closest to the area of hexagon AA′CC′BB′AA'CC'BB'.
11.
The integers from 1 to 64 are placed in some order into an 8×88\times8 grid of cells with one number in each cell. Let ai,ja_{i,j} be the number placed in the cell in row ii and column jj, and let MM be the sum of the absolute differences between adjacent cells. That is,
M=∑i=18∑j=17(∣ai,j+1−ai,j∣+∣aj+1,i−aj,i∣).M=\sum_{i=1}^{8}\sum_{j=1}^{7}(|a_{i,j+1}-a_{i,j}|+|a_{j+1,i}-a_{j,i}|).
Find the remainder when the maximum possible value of MM is divided by 10001000.
12.
Triangle △ABC\triangle ABC lies in plane D\mathcal{D} with AB=6AB=6, AC=4AC=4, and ∠BAC=90∘\angle BAC=90^\circ. Let DD be the reflection across BCBC of the centroid of △ABC\triangle ABC. For four spheres, all on the same side of P\mathcal{P}, have radii 1,2,3,1,2,3, and rr and are tangent to D\mathcal{D} at points A,B,C,A,B,C, and DD, respectively. The four spheres are also each tangent to a second plane T\mathcal{T} and are all on the same side of T\mathcal{T}. The value of rr can be written as mn\frac{m}{n}, where mm and nn are relatively prime positive integers. Find m+nm+n.
13.
For each nonnegative integer rr less than 502, define
Sr=∑m≥0(10000502m+r),S_r=\sum_{m\ge0}\binom{10000}{502m+r},
where (10000n)\binom{10000}{n} is defined to be 0 when n>10,000n>10,000. That is, SrS_r is the sum of all binomial coefficients of the form (10000k)\binom{10000}{k} for which 0≤k≤10,0000\le k\le10,000 and k−rk-r is a multiple of 502. Find the number of integers in the list S0,S1,S2,…,S501S_0,S_1,S_2,\ldots,S_{501} that are multiples of the prime number 503.
14.
In an equiangular pentagon, the sum of the squares of the side lengths equals 308, and the sum of the squares of the diagonal lengths equals 800. The square of the perimeter of the pentagon can be expressed as mnm\sqrt n, where mm and nn are positive integers and nn is not divisible by the square of any prime. Find m+nm+n.
15.
Let aa, bb, and nn be positive integers with both aa and bb greater than or equal to 22 and less than or equal to 2n2n. Define an a×ba\times b cell loop in a 2n×2n2n\times2n grid of cells to be the 2a+2b−42a+2b-4 cells that surround an (a−2)×(b−2)(a-2)\times(b-2) (possibly empty) rectangle of cells in the grid. For example, the following diagram shows a way to partition a 6×66\times6 grid of cells into 44 cell loops.

Find the number of ways to partition a 10×1010\times10 grid of cells into 55 cell loops so that every cell of the grid belongs to exactly one cell loop.
Example partition of a 6 by 6 grid into 4 cell loops