Question 160
Find all pairs $(a, b)$ of positive integers such that $f(x) = x$ is the only function $f: \mathbb{R} \to \mathbb{R}$ that satisfies
$$
f^a(x)f^b(y) + f^b(x)f^a(y) = 2xy
$$
for all $x, y \in \mathbb{R}$.
Here $f^n(x)$ represents the function obtained by applying $n$ times the function $f$ to $x$, so $f^1(x) = f(x)$ and $f^{n+1}(x) = f(f^n(x))$.
$$
f^a(x)f^b(y) + f^b(x)f^a(y) = 2xy
$$
for all $x, y \in \mathbb{R}$.
Here $f^n(x)$ represents the function obtained by applying $n$ times the function $f$ to $x$, so $f^1(x) = f(x)$ and $f^{n+1}(x) = f(f^n(x))$.