USAJMO 2025

Let $\mathbb Z$ be the set of integers, and let $f\colon \mathbb Z \to \mathbb Z$ be a function. Prove that there are infinitely many integers $c$ such that the function $g\colon \mathbb Z \to \mathbb Z$ defined by $g(x) = f(x) + cx$ is not bijective. Note: A function $g\colon \mathbb Z \to \mathbb Z$ is bijective if for every integer $b$ , there exists exactly one integer $a$ such that $g(a) = b$ .

Let $k$ and $d$ be positive integers. Prove that there exists a positive integer $N$ such that for every odd integer $n>N$ , the digits in the base- $2n$ representation of $n^k$ are all greater than $d$ .

Let $m$ and $n$ be positive integers, and let $\mathcal R$ be a $2m\times{2n}$ grid of unit squares.

A domino is a $1\times2{}$ or $2\times{1}$ rectangle. A subset $S$ of grid squares in $\mathcal R$ is domino-tileable if dominoes can be placed to cover every square of $S$ exactly once with no domino extending outside of $S$ . Note: The empty set is domino tileable.

An up-right path is a path from the lower-left corner of $\mathcal R$ to the upper-right corner of $\mathcal R$ formed by exactly $2m+2n$ edges of the grid squares.

Determine, with proof, in terms of $m$ and $n$ , the number of up-right paths that divide $\mathcal R$ into two domino-tileable subsets.

Let $n$ be a positive integer, and let $a_0, a_1,\dots, a_n$ be nonnegative integers such that $a_0\ge a_1\ge \dots\ge a_n.$ Prove that $$\sum_{i=0}^n i\binom{a_i}{2}\le\frac{1}{2}\binom{a_0+a_1+\dots+a_n}{2}.$$ Note: $\binom{k}{2}=\frac{k(k-1)}{2}$ for all nonnegative integers $k$ .

Let $H$ be the orthocenter of acute triangle $ABC$ , let $F$ be the foot of the altitude from $C$ to $AB$ , and let $P$ be the reflection of $H$ across $BC$ . Suppose that the circumcircle of triangle $AFP$ intersects line $BC$ at two distinct points $X$ and $Y$ . Prove that $C$ is the midpoint of $XY$ .

Let $S$ be a set of integers with the following properties:

• $\{ 1, 2, \dots, 2025 \} \subseteq S$ .
• If $a, b \in S$ and $\gcd(a, b) = 1$ , then $ab \in S$ .
• If for some $s \in S$ , $s + 1$ is composite, then all positive divisors of $s + 1$ are in $S$ .

Prove that $S$ contains all positive integers.