Problem

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

Tags: algebra



Does there exist a positive integer $n$ and a) complex numbers $a_0, a_1, \ldots, a_n;$ b) reals $a_0, a_1, \ldots, a_n, $ such that $P(x) Q(x)=x^{2024}+1$ where $P(x)=a_nx^n+\ldots +a_1x+a_0$ and $Q(x)=a_0x^n+a_1x^{n-1}+\ldots+a_n?$