ISL 2006 N4

Number theory
Difficulty band 6

Let $P$ be a polynomial of degree $n>1$ with integer coefficients and let $k$ be any positive integer. Consider the polynomial $Q(x)=P(P(\ldots P(P(x)) \ldots))$ , with $k$ pairs of parentheses. Prove that $Q$ has no more than $n$ integer fixed points, i.e. integers satisfying the equation $Q(x)=x$ . (Romania)

Official text Transcription from NuminaMath olympiads, checked word for word against the official paper. Quoted with source.

Official text: IMO 2006 Shortlist booklet, page 59.

Solutions

Official solution Solution 1 by the IMO Problem Selection Committee

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