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Polynomial equation implies real root

Source: 2022 Bulgarian Spring Math Competition, Problem 12.3

March 27, 2022
polynomialalgebra

Problem Statement

Let P,Q∈R[x]P,Q\in\mathbb{R}[x], such that QQ is a 20212021-degree polynomial and let a1,a2,…,a2022,b1,b2,…,b2022a_{1}, a_{2}, \ldots , a_{2022}, b_{1}, b_{2}, \ldots , b_{2022} be real numbers such that a1a2…a2022≠0a_{1}a_{2}\ldots a_{2022}\neq 0. If for all real xx P(a1Q(x)+b1)+…+P(a2021Q(x)+b2021)=P(a2022Q(x)+b2022)P(a_{1}Q(x) + b_{1}) + \ldots + P(a_{2021}Q(x) + b_{2021}) = P(a_{2022}Q(x) + b_{2022}) prove that P(x)P(x) has a real root.
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