IMO 2004 — Complete Shortlist Question 26

A function $f$ from the set of positive integers $\mathbb{N}$ into itself is such that for all $m, n \in \mathbb{N}$ the number $(m^2 + n)^2$ is divisible by $f^2(m) + f(n)$. Prove that $f(n) = n$ for each $n \in \mathbb{N}$.