Question 6
Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that
$$
f^{a^{2}+b^{2}}(a+b)=a f(a)+b f(b) \quad \text { for every } a, b \in \mathbb{Z}
$$
Here, $f^{n}$ denotes the $n^{\text {th }}$ iteration of $f$, i.e., $f^{0}(x)=x$ and $f^{n+1}(x)=f\left(f^{n}(x)\right)$ for all $n \geqslant 0$.
$$
f^{a^{2}+b^{2}}(a+b)=a f(a)+b f(b) \quad \text { for every } a, b \in \mathbb{Z}
$$
Here, $f^{n}$ denotes the $n^{\text {th }}$ iteration of $f$, i.e., $f^{0}(x)=x$ and $f^{n+1}(x)=f\left(f^{n}(x)\right)$ for all $n \geqslant 0$.