Školjka

%V0 Find all functions $f: \mathbb{N^{*}}\to \mathbb{N^{*}}$ satisfying $$\left(f^{2}\left(m\right)+f\left(n\right)\right) \mid \left(m^{2}+n\right)^{2}$$ for any two positive integers $m$ and $n$. Remark. The abbreviation $\mathbb{N^{*}}$ stands for the set of all positive integers: $\mathbb{N^{*}}=\left\{1,2,3,...\right\}$. By $f^{2}\left(m\right)$, we mean $\left(f\left(m\right)\right)^{2}$ (and not $f\left(f\left(m\right)\right)$).