Question bank AIME 2026

Find the sum of the $10$th terms of all arithmetic sequences of integers that have first term equal to $4$ and include both $24$ and $34$ as terms.

The figure below shows a grid of $10$ squares in a row. Each square has a diagonal connecting its lower left vertex to its upper right vertex. A bug moves along the line segments from vertex to vertex, never traversing the same segment twice and never moving from right to left along a horizontal or diagonal segment. Let $N$ be the number of paths the bug can take from the lower left corner $(A)$ to the upper right corner $(B).$ One such path from $A$ to $B$ is shown by the thick line segments in the figure. Find $\sqrt{N}.$

Let $ABCDE$ be a nonconvex pentagon with internal angles $\angle A = \angle E = 90^\circ$ and $\angle B = \angle D = 45^\circ.$ Suppose that $DE \lt AB,$ $AE = 20,$ $BC = 14\sqrt{2},$ and points $B,$ $C,$ and $D$ lie on the same side of line $AE.$ Suppose further that $AB$ is an integer with $AB \lt 2026$ and the area of pentagon $ABCDE$ is an integer multiple of $16.$ Find the number of possible values of $AB.$

For each positive integer $n$ let $f(n)$ be the value of the base-ten numeral $n$ viewed in base $b,$ where $b$ is the least integer greater than the greatest digit in $n.$ For example, if $n = 72,$ then $b = 8,$ and $72$ as a numeral in base $8$ equals $7 \cdot 8 + 2 = 58;$ therefore $f(72) = 58.$ Find the number of positive integers $n$ less than $1000$ such that $f(n) = n.$

An urn contains $n$ marbles. Each marble is either red or blue, and there are at least $7$ marbles of each color. When $7$ marbles are drawn randomly from the urn without replacement, the probability that exactly $4$ of them are red equals the probability that exactly $5$ of them are red. Find the sum of the five least values of $n$ for which this is possible.

Find the sum of all real numbers $r$ such that there is at least one point where the circle with radius $r$ centered at $(4, 39)$ is tangent to the parabola with equation $2y = x^2 - 8x + 12.$

A standard fair six-sided die is rolled repeatedly. Each time the die reads $1$ or $2,$ Alice gets a coin; each time it reads $3$ or $4,$ Bob gets a coin; and each time it reads $5$ or $6,$ Carol gets a coin. The probability that Alice and Bob each receive at least two coins before Carol receives any coins can be written as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $100m + n.$

Isosceles triangle $\triangle ABC$ has $AB = BC.$ Let $I$ be the incenter of $\triangle ABC.$ The perimeters of $\triangle ABC$ and $\triangle AIC$ are in the ratio $125 : 6,$ and all the sides of both triangles have integer lengths. Find the minimum possible value of $AB.$

Let $S$ denote the value of the infinite sum $$
\frac{1}{9} + \frac{1}{99} + \frac{1}{999} + \frac{1}{9999} + \cdots
$$ Find the remainder when the greatest integer less than or equal to $10^{100} S$ is divided by $1000.$

Let $\triangle ABC$ be a triangle with $D$ on $\overline{BC}$ such that $\overline{AD}$ bisects $\angle BAC.$ Let $\omega$ be the circle that passes through $A$ and is tangent to segment $\overline{BC}$ at $D.$ Let $E \ne A$ and $F \ne A$ be the intersections of $\omega$ with segments $\overline{AB}$ and $\overline{AC},$ respectively. Suppose that $AB = 200,$ $AC = 225,$ and all of $AE,$ $AF,$ $BD,$ and $CD$ are positive integers. Find the greatest possible value of $BC.$

Find the greatest integer $n$ such that the cubic polynomial $$
x^3 - \frac{n}{6}x^2 + (n - 11)x - 400
$$ has roots $\alpha^2,$ $\beta^2,$ and $\gamma^2,$ where $\alpha,$ $\beta,$ and $\gamma$ are complex numbers, and there are exactly seven different possible values for $\alpha + \beta + \gamma.$

Consider a tetrahedron with two isosceles triangle faces with side lengths $5\sqrt{10},$ $5\sqrt{10},$ and $10$ and two isosceles triangle faces with side lengths $5\sqrt{10},$ $5\sqrt{10},$ and $18.$ The four vertices of the tetrahedron lie on a sphere with center $S,$ and the four faces of the tetrahedron are tangent to a sphere with center $R.$ The distance $RS$ can be written as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Call finite sets of integers $S$ and $T$ *cousins* if

• $S$ and $T$ have the same number of elements,

• $S$ and $T$ are disjoint, and

• the elements of $S$ can be paired with the elements of $T$ so that the elements in each pair differ by exactly $1.$

For example, $\{1, 2, 5\}$ and $\{0, 3, 4\}$ are cousins. Suppose that the set $S$ has exactly $4040$ cousins. Find the least number of elements the set $S$ can have.

For integers $a$ and $b,$ let $a \circ b = a - b$ if $a$ is odd and $b$ is even, and $a \circ b = a + b$ otherwise. Find the number of sequences $a_1,$ $a_2,$ $a_3,$ $\ldots,$ $a_n$ of positive integers such that $$
a_1 + a_2 + a_3 + \cdots + a_n = 12
$$ and $$
a_1 \circ a_2 \circ a_3 \circ \cdots \circ a_n = 0,
$$ where the operations are performed from left to right; that is, $a_1 \circ a_2 \circ a_3$ means $(a_1 \circ a_2) \circ a_3.$

Find the number of ordered $7$-tuples $(a_1, a_2, a_3, \ldots, a_7)$ having the following properties:

• $a_k \in \{1, 2, 3\}$ for all $k.$

• $a_1 + a_2 + a_3 + a_4$ $+ a_5 + a_6 + a_7$ is a multiple of $3.$

• $a_1a_2a_4$ $+ a_2a_3a_5$ $+ a_3a_4a_6$ $+ a_4a_5a_7$ $+ a_5a_6a_1$ $+ a_6a_7a_2$ $+ a_7a_1a_3$ is a multiple of $3.$

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, José 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 $\frac{m}{n}$ miles, where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

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.

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 $\mathcal{T}$ be the region of points $P$ in the disk such that a sphere of radius $42$ can be placed on top of the disk at $P$ and lie completely inside the hemisphere. The area of $\mathcal{T}$ divided by the area of the disk is $\frac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find $p + q.$

Find the number of integers less than or equal to $100$ that are equal to $a + b + ab$ for some choice of distinct positive integers $a$ and $b.$

A plane contains points $A$ and $B$ with $AB = 1.$ Point $A$ is rotated in the plane counterclockwise through an acute angle $\theta$ around point $B$ to point $A'.$ Then $B$ is rotated in the plane clockwise through angle $\theta$ around point $A'$ to point $B'.$ Suppose $AB' = \frac{4}{3}.$ The value of $\cos\theta$ can be written as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$