Question bank USAJMO 2024

Let $ABCD$ be cyclic, with $AB=7$ and $CD=8$. Points $P,Q$ lie on segment $AB$ with $AP=BQ=3$. Points $R,S$ lie on segment $CD$ with $CR=DS=2$. Prove that $PQRS$ is cyclic.

Proposed by Evan o’Dorney. Source: Evan Chen’s third-party compilation; https://aops.com/community/p30216434. One editorial point; manual grading.

Let $m,n$ be positive integers and let $S$ be the lattice points $(x,y)$ with $1\le x\le2m$ and $1\le y\le2n$. A configuration of $mn$ axis-parallel rectangles is happy if every point of $S$ is a vertex of exactly one rectangle. Prove that the number of happy configurations is odd.

Proposed by Serena An and Claire Zhang. Source: Evan Chen’s third-party compilation; https://aops.com/community/p30216444. One editorial point; manual grading.

Positive integers $a_1,a_2,\ldots$ satisfy $a_1=2$ and

$$a_{n+1}=a_n^{n+1}-1\qquad(n\ge1).$$

Prove that for every odd prime $p$ and integer $k$, some term is divisible by $p^k$.

Proposed by John Berman. Source: Evan Chen’s third-party compilation; https://aops.com/community/p30216450. One editorial point; manual grading.

Let $n\ge3$. Rowan may permute rows and Colin may permute columns of an $n\times n$ grid colored red or blue. A coloring is orderly if every row permutation can be undone by a column permutation, and every column permutation can be undone by a row permutation, restoring the original coloring in each case. How many orderly colorings are there?

Proposed by Alec Sun. Source: Evan Chen’s third-party compilation; https://aops.com/community/p30227193. One editorial point; manual grading.

Find all functions $f:\mathbb R\to\mathbb R$ such that, for all real $x,y$,

$$f(x^2-y)+2yf(x)=f(f(x))+f(y).$$

Proposed by Carl Schildkraut. Source: Evan Chen’s third-party compilation; https://aops.com/community/p30227204. One editorial point; manual grading.

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. One nonofficial editorial point; manual essay grading.