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Functional equation with polynomial component

Source: Vietnam TST 2024 P1

March 27, 2024
algebrafunctional equationpolynomialfunction

Problem Statement

Let P(x)∈R[x]P(x) \in \mathbb{R}[x] be a monic, non-constant polynomial. Determine all continuous functions f:R→Rf: \mathbb{R} \to \mathbb{R} such that f(f(P(x))+y+2023f(y))=P(x)+2024f(y),f(f(P(x))+y+2023f(y))=P(x)+2024f(y), for all reals x,yx,y.
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