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Factoring a composition of two polynomials

Source: Czech-Polish-Slovak Match, 2011

August 9, 2011
algebrapolynomialalgebra unsolved

Problem Statement

A polynomial P(x)P(x) with integer coefficients satisfies the following: if F(x)F(x), G(x)G(x), and Q(x)Q(x) are polynomials with integer coefficients satisfying P(Q(x))=F(x)ā‹…G(x)P\Big(Q(x)\Big)=F(x)\cdot G(x), then F(x)F(x) or G(x)G(x) is a constant polynomial. Prove that P(x)P(x) is a constant polynomial.
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