Question 9
Find all pairs $(a, b)$ of positive integers such that $f(x)=x$ is the only function $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfies
$$
f^{a}(x) f^{b}(y)+f^{b}(x) f^{a}(y)=2 x y
$$
for all $x, y \in \mathbb{R}$.
Here $f^{n}(x)$ denotes applying $f$ to $x$ $n$ times, so $f^{1}(x)=f(x)$ and $f^{n+1}(x)=f\left(f^{n}(x)\right)$.
$$
f^{a}(x) f^{b}(y)+f^{b}(x) f^{a}(y)=2 x y
$$
for all $x, y \in \mathbb{R}$.
Here $f^{n}(x)$ denotes applying $f$ to $x$ $n$ times, so $f^{1}(x)=f(x)$ and $f^{n+1}(x)=f\left(f^{n}(x)\right)$.