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Functional polynomial equation

Source: Mexico National Olympiad Mock Exam 2021 P1

November 12, 2021
functionalgebrapolynomial

Problem Statement

Find all functions f ⁣:R→Rf \colon \mathbb{R} \to \mathbb{R} that satisfy the following property for all real numbers xx and all polynomials PP with real coefficients:
If P(f(x))=0P(f(x)) = 0, then f(P(x))=0f(P(x)) = 0.
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