International Mathematical Olympiad 2010 Problem 3

Let $\mathbb{N}$ be the set of positive integers. Determine all functions $g\colon \mathbb{N}\to \mathbb{N}$ such that $$\left(g (m) + n\right) \left(m + g (n)\right)$$ is a perfect square for all $m, n \in \mathbb{N}$.