ELMO SL 2012 N8

Fix two positive integers $a,k\ge2$ , and let $f\in\mathbb{Z}[x]$ be a nonconstant polynomial. Suppose that for all sufficiently large positive integers $n$ , there exists a rational number $x$ satisfying $f(x)=f(a^n)^k$ . Prove that there exists a polynomial $g\in\mathbb{Q}[x]$ such that $f(g(x))=f(x)^k$ for all real $x$ .

Victor Wang.