RO-TST 2005 Problem 16

Let $\mathbb{N}=\{1,2,\ldots\}$ . Find all functions $f: \mathbb{N}\to\mathbb{N}$ such that for all $m,n\in \mathbb{N}$ the number $f^2(m)+f(n)$ is a divisor of $(m^2+n)^2$ .