CMO 2002 Problem 5

Let $\mathbb{N}=\{0,1,2, \ldots\}$. Determine all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that

$$x f(y)+y f(x)=(x+y) f\left(x^{2}+y^{2}\right)$$

for all $x$ and $y$ in $\mathbb{N}$.