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11S Three ways to attack a functional equation

11. PEN K11 (Canada 2002) Find all functions $f: \mathbb{N}_{0}\to \mathbb{N}_{0}$ such that for all

$m, \, n\in \mathbb{N}_{0}$:

$mf(n)+nf(m)=(m+n)f(m^{2}+n^{2}).$

Here, $\mathbb{N}_{0}$ denote the set of all nonnegative integers.

Here is the official solution file: pen11.pdf.
You can also discuss the problems here!

11. Three ways to attack a functional equation

Hi everyone. PEN is BACK. Here goes the first problem of the second season!

11. PEN K11 (Canada 2002) Find all functions $f: \mathbb{N}_{0}\to \mathbb{N}_{0}$ such that for all
$m, \, n\in \mathbb{N}_{0}$:

$mf(n)+nf(m)=(m+n)f(m^{2}+n^{2}).$

Here, $\mathbb{N}_{0}$ denote the set of all nonnegative integers.

This problem is suggested by Alexander Remorov from Canada. Next week, his solution will be posted!