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Two polynomials, prove P(P(x))=Q(Q(x)) has no real solution

Source: Canadian Mathematical Olympiad - 1981 - Problem 4.

May 27, 2011
algebrapolynomialfunctionalgebra proposed

Problem Statement

P(x),Q(x)P(x),Q(x) are two polynomials such that P(x)=Q(x)P(x)=Q(x) has no real solution, and P(Q(x))≡Q(P(x))∀x∈RP(Q(x))\equiv Q(P(x))\forall x\in\mathbb{R}. Prove that P(P(x))=Q(Q(x))P(P(x))=Q(Q(x)) has no real solution.
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