AIME 2026b

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 $\sqrt{N}$. [asy] size(12cm); draw((0, 0)--(0, 1)); for (int i = 0; i <= 9; i += 1) { draw((i, 1)--(i + 1, 1)--(i + 1, 0)--(i, 0)--(i + 1, 1)); } draw((0, 0)--(0, 1)--(2, 1)--(2, 0)--(3, 1)--(3, 0)--(4, 1)--(4, 0)--(5, 0)--(5, 1)--(6, 1)--(6, 0)--(7, 1)--(8, 1)--(8, 0)--(10, 0)--(10, 1), red + linewidth(2)); label("$A$", (0, 0), dir(180)); label("$B$", (10, 1), dir(0)); [/asy]

Let $ABCDE$ be a nonconvex pentagon with internal angles $\angle A = \angle E = 90^\circ$ and $\angle B = \angle D = 45^\circ$. Suppose that $DE < AB$, $AE = 20$, $BC = 14\sqrt2$, and points $B$, $C$, and $D$ lie on the same side of line $AE$. Suppose further that $AB$ is an integer with $AB < 2026$ and the area of pentagon $ABCDE$ is an integer multiple of $16$. Find the number of possible values of $AB$. [asy] size(7cm); import geometry; pair A = (0,0); pair E = (20,0); pair B = (0, 17); pair D = (20, 9); pair C = (14, 3); draw(A--B--C--D--E--cycle); draw(A--E, dashed); dot(A); dot(B); dot(C); dot(D); dot(E); label("A", A, SW); label("B", B, W); label("C", C, N); label("D", D, N); label("E", E, SE); markrightangle(E,A,B); markrightangle(D,E,A); markangle(A,B,C, radius=10); markangle(C,D,E, radius=10); label("$20$", midpoint(A--E), S); label("$14\sqrt{2}$", midpoint(B--C), NE);[/asy] ~Diagram by sillybone

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 $\tfrac mn,$ 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 $$\frac{1}{9}+\frac{1}{99}+\frac{1}{999}+\frac{1}{9999}+\dots.$$ 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 $E \neq A$ and $F \neq A$ 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 $\tfrac mn,$ where $m$ and $n$ are relatively prime positive integers. Find ${}m + n.$ [asy] size(320); pair S = (0, 0); pair A = (0, 110); pair D = (-95, -55); pair C = (95, -55); pair B = (-25, 15); pair R = (0, 25); draw(circle(S, 110), gray+linewidth(0.6)); draw(circle(R, 43), gray+linewidth(0.4)); draw(A--C, black+linewidth(1.1)); draw(A--D, black+linewidth(1.1)); draw(C--D, black+linewidth(1.1)); draw(A--B, dashed+gray+linewidth(0.9)); draw(B--C, dashed+gray+linewidth(0.9)); draw(B--D, dashed+gray+linewidth(0.9)); draw(S--R, dotted+black+linewidth(1.3)); dot(A, p=black+linewidth(4.5)); dot(B, p=black+linewidth(4.5)); dot(C, p=black+linewidth(4.5)); dot(D, p=black+linewidth(4.5)); dot(S, p=red+linewidth(4.5)); dot(R, p=blue+linewidth(4.5)); label("$A$", A, N3.5); label("$B$", B, NW3.0); label("$C$", C, SE3.5); label("$D$", D, SW3.5); label("$S$", S, SW2.0); label("$R$", R, NE2.0); label("$10$", (A+B)/2, NE2.5); label("$10$", (C+D)/2, S3.0); label("$18$", (A+D)/2, NW3.0); label("$18$", (B+C)/2, N2.5); label("$5\sqrt{10}$", (A+C)/2, NE3.0); label("$5\sqrt{10}$", (B+D)/2, S3.0); [/asy]

Call finite sets of integers $S$ and $T$ cousins if 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 $a+b$ otherwise. Find the number of sequences $a_1,a_2,a_3,\dots,a_n$ of positive integers such that $$a_1+a_2+a_3+\dots + a_n = 12\text{ and }a_1\circ a_2\circ a_3\circ\dots\circ a_n = 0,$$ 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: