RMM 2010 Problem 6

Given a polynomial $f(x)$ with rational coefficients, of degree $d \ge 2$ , we define the sequence of sets $f^0(\mathbb{Q}), f^1(\mathbb{Q}), \ldots$ as $f^0(\mathbb{Q})=\mathbb{Q}$ , $f^{n+1}(\mathbb{Q})=f(f^{n}(\mathbb{Q}))$ for $n\ge 0$ . (Given a set $S$ , we write $f(S)$ for the set $\{f(x)\mid x\in S\})$ . Let $f^{\omega}(\mathbb{Q})=\bigcap_{n=0}^{\infty} f^n(\mathbb{Q})$ be the set of numbers that are in all of the sets $f^n(\mathbb{Q})$ , $n\geq 0$ . Prove that $f^{\omega}(\mathbb{Q})$ is a finite set.

Dan Schwarz, Romania