ELMO 2025 Problem 1

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