2026 AIME II

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 $3.$