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P(x)f(Q(x))=f(P(x)Q(x)) if f(x)+f(y)=f(x+y)

Source: 2021 IMOC qualification problem, A2

December 30, 2021
algebrapolynomialfunctional equationfunctional

Problem Statement

Find all functions f:Rā†’Rf:R \to R, such that f(x)+f(y)=f(x+y)f(x)+f(y)=f(x+y), and there exists non-constant polynomials P(x)P(x), Q(x)Q(x) such that P(x)f(Q(x))=f(P(x)Q(x))P(x)f(Q(x))=f(P(x)Q(x))
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