Question 57
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 $\mathbb{N}$.
$$
x f(y) + y f(x) = (x + y) f\left(x^{2} + y^{2}\right)
$$
for all $x$ and $y$ in $\mathbb{N}$.