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cubic polynomial and sequence

Source: IMO ShortList 1990, Problem 26 (USA 2)

April 9, 2005
algebrapolynomialnumber theoryCubicIterationIMO Shortlist

Problem Statement

Let p(x) p(x) be a cubic polynomial with rational coefficients. q1 q_1, q2 q_2, q3 q_3, ... is a sequence of rationals such that q_n \equal{} p(q_{n \plus{} 1}) for all positive n n. Show that for some k k, we have q_{n \plus{} k} \equal{} q_n for all positive n n.
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