BALKAN 2009 Problem 4

Determine all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ satisfying

$$f\left(f(m)^{2}+2 f(n)^{2}\right)=m^{2}+2 n^{2} \text { for all } m, n \in \mathbb{N} . \text { (Bulgaria) }$$

Time allowed: 270 minutes. Each problem is worth 10 points.