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$r_n = p(r_{n+1} ) $, p is a polynomial

Source: 13-th Hungary-Israel Binational Mathematical Competition 2002

April 7, 2007
algebrapolynomialalgebra unsolved

Problem Statement

Let p(x)p(x) be a polynomial with rational coefficients, of degree at least 22. Suppose that a sequence (rn)(r_{n}) of rational numbers satisfies rn=p(rn+1)r_{n}= p(r_{n+1}) for every n≥1n\geq 1. Prove that the sequence (rn)(r_{n}) is periodic.
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