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Sth simple but not simple

Source: Thailand TSTST 2024 P12

July 18, 2024
algebrapolynomial

Problem Statement

A polynomial A(x)A(x) is said to be simple if A(x)A(x) is divisible by xx but not divisible by x2x^2. Suppose that a polynomial P(x)P(x) has a simple polynomial Q(x)Q(x) such that P(Q(x))−Q(2x)P(Q(x))-Q(2x) is divisible by x2x^2. Prove that there exists a simple polynomial R(x)R(x) such that P(R(x))−R(2x)P(R(x))-R(2x) is divisible by x2023x^{2023}.
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