USAJMO 2024

Let $ABCD$ be a cyclic quadrilateral with $AB = 7$ and $CD = 8$ . Point $P$ and $Q$ are selected on segment $AB$ such that $AP = BQ = 3$ . Points $R$ and $S$ are selected on segment $CD$ such that $CR = DS = 2$ . Prove that $PQRS$ is a cyclic quadrilateral.

Proposed by Evan O'Dorney

Let $m$ and $n$ be positive integers. Let $S$ be the set of integer points $(x,y)$ with $1\leq x\leq 2m$ and $1\leq y\leq 2n$ . A configuration of $mn$ rectangles is called happy if each point in $S$ is a vertex of exactly one rectangle, and all rectangles have sides parallel to the coordinate axes. Prove that the number of happy configurations is odd.

Proposed by Serena An and Claire Zhang

A sequence $a_{1}, a_{2}, \ldots$ of positive integers is defined recursively by $a_{1}=2$ and $$a_{n+1}=a_{n}^{n+1}-1 \text { for } n \geq 1$$ Prove that for every odd prime $p$ and integer $k$, some term of the sequence is divisible by $p^{k}$.

Let $n \geq 3$ be an integer. Rowan and Colin play a game on an $n \times n$ grid of squares, where each square is colored either red or blue. Rowan is allowed to permute the rows of the grid and Colin is allowed to permute the columns. A grid coloring is orderly if:

• no matter how Rowan permutes the rows of the coloring, Colin can then permute the columns to restore the original grid coloring; and
• no matter how Colin permutes the columns of the coloring, Rowan can then permute the rows to restore the original grid coloring. In terms of $n$ , how many orderly colorings are there?

Proposed by Alec Sun

Point $D$ is selected inside acute $\triangle ABC$ so that $\angle DAC = \angle ACB$ and $\angle BDC = 90^{\circ} + \angle BAC$ . Point $E$ is chosen on ray $BD$ so that $AE = EC$ . Let $M$ be the midpoint of $BC$ .

Show that line $AB$ is tangent to the circumcircle of triangle $BEM$ .

Proposed by Anton Trygub