ELMO 2025

Let $\mathbb{Z}_{>0}$ denote the set of positive integers. Find all functions $f \colon \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ such that for all positive integers $m$ and $n$ , $$f^m(n)+f(mn)=f(m)f(n).$$

Note: $f^m(n)=\underbrace{f(f(\cdots f}_{m\text{ times}}(n)\cdots))$ , that is, $f$ applied $m$ times to $n$ .

Seongjin Shim

A contest has $2025$ problems. Elmo initially knows that the answers are $1$ , $2$ , $\dots$ , $2025$ in some order. Every minute, Elmo selects a problem that has not been answered correctly, inputs an answer to it, and is told whether the answer is correct or incorrect. Elmo wins the contest if and only if he answers all $2025$ problems correctly in at most $2024$ attempts each. Can Elmo guarantee that he wins the contest?

Andrew Lin and Calvin Wang

Let $n$ be a positive integer and $p$ be a prime. In terms of $n$ and $p$ , find the largest nonnegative integer $k$ for which there exists a polynomial $P(x)$ with integer coefficients satisfying the following conditions:

• The $x^n$ coefficient of $P(x)$ is $1$ .
• $p^k$ divides $P(x)$ for all integers $x$ .

Aprameya Tripathy

In quadrilateral $ABCD$ with $\angle BAD=\angle BCD$ , diagonal $AC$ intersects the circumcircle of $\triangle BCD$ at a point $P \neq C$ . Let $Q$ , $R$ , $S$ and $T$ be the reflections of $P$ across $\overline{AB}$ , $\overline{BC}$ , $\overline{CD}$ , and $\overline{DA}$ , respectively. Prove that the circumcircle of $\triangle AQT$ is tangent to line $RS$ .

Benny Wang

Let $k$ be a positive integer. Define a sequence of positive integers $a_1$ , $a_2$ , $\dots$ by $a_1=1$ and $$a_{n+1}=a_n+k^{a_n}$$for all positive integers $n$ . Show that there exist infinitely many primes $p$ such that for any integer $r$ , there exists an index $m$ such that $a_m\equiv r\pmod p$ .

Oron Wang

Let $n \ge 2$ be an integer. An $n \times n$ grid is filled with the numbers $1$ through $n$ such that each row contains every number exactly once and any two adjacent rows differ by exactly two elements. Find all $n$ such that for any valid labeling of the grid, there exists a set of $n$ cells, each with a different number, such that no two are on the same row or column.

Andrew Carratu