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Product of a polynomial and its inverse is x^{2024}+1

Source: Serbia Additional IMO TST 2024, P1 (out of 4)

May 30, 2024
algebra

Problem Statement

Does there exist a positive integer nn and a) complex numbers a0,a1,…,an;a_0, a_1, \ldots, a_n; b) reals a0,a1,…,an,a_0, a_1, \ldots, a_n, such that P(x)Q(x)=x2024+1P(x) Q(x)=x^{2024}+1 where P(x)=anxn+…+a1x+a0P(x)=a_nx^n+\ldots +a_1x+a_0 and Q(x)=a0xn+a1xn−1+…+an?Q(x)=a_0x^n+a_1x^{n-1}+\ldots+a_n?
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