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Divisibility of polynomials

Source: Polish MO second round 2011

February 19, 2012
algebrapolynomialcalculusintegrationalgebra unsolved

Problem Statement

There are two given different polynomials P(x),Q(x)P(x),Q(x) with real coefficients such that P(Q(x))=Q(P(x))P(Q(x))=Q(P(x)). Prove that ∀n∈Z+\forall n\in \mathbb{Z_{+}} polynomial: P(P(…P(P⏟n(x))…))−Q(Q(…Q(Q⏟n(x))…))\underbrace{P(P(\ldots P(P}_{n}(x))\ldots))- \underbrace{Q(Q(\ldots Q(Q}_{n}(x))\ldots)) is divisible by P(x)−Q(x)P(x)-Q(x).
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