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Romanian Masters in mathematics 2010 Day 2 Problem 3

Source:

April 25, 2010
algebrapolynomialinductionpigeonhole principlenumber theoryrelatively prime

Problem Statement

Given a polynomial f(x)f(x) with rational coefficients, of degree d≥2d \ge 2, we define the sequence of sets f0(Q),f1(Q),…f^0(\mathbb{Q}), f^1(\mathbb{Q}), \ldots as f0(Q)=Qf^0(\mathbb{Q})=\mathbb{Q}, fn+1(Q)=f(fn(Q))f^{n+1}(\mathbb{Q})=f(f^{n}(\mathbb{Q})) for n≥0n\ge 0. (Given a set SS, we write f(S)f(S) for the set {f(x)∣x∈S})\{f(x)\mid x\in S\}). Let fω(Q)=⋂n=0∞fn(Q)f^{\omega}(\mathbb{Q})=\bigcap_{n=0}^{\infty} f^n(\mathbb{Q}) be the set of numbers that are in all of the sets fn(Q)f^n(\mathbb{Q}), n≥0n\geq 0. Prove that fω(Q)f^{\omega}(\mathbb{Q}) is a finite set.
Dan Schwarz, Romania
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