Question 32
Given a polynomial $f(x)$ with rational coefficients, of degree $d \ge 2$, we define the sequence of sets $f^0(\mathbb{Q}), f^1(\mathbb{Q}), \dots$, by $f^0(\mathbb{Q}) = \mathbb{Q}$ and then $f^{n+1}(\mathbb{Q}) = f(f^n(\mathbb{Q}))$ 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 $f^n(\mathbb{Q})$. Prove that $f^\omega(\mathbb{Q})$ is a finite set.
(Romania) Dan Schwarz
(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 $f^n(\mathbb{Q})$. Prove that $f^\omega(\mathbb{Q})$ is a finite set.
(Romania) Dan Schwarz