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Old idea in a new context

Source: Vietnam TST 2024 P6

March 27, 2024
algebrapolynomial

Problem Statement

Let P(x)∈Z[x]P(x) \in \mathbb{Z}[x] be a polynomial. Determine all polynomials Q(x)∈Z[x]Q(x) \in \mathbb{Z}[x], such that for every positive integer nn, there exists a polynomial Rn(x)∈Z[x]R_n(x) \in \mathbb{Z}[x] satisfies Q(x)2n−1=Rn(x)(P(x)2n−1).Q(x)^{2n} - 1 = R_n(x)\left(P(x)^{2n} - 1\right).
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