ELMO 2022 Problem 2

Find all monic nonconstant polynomials $P$ with integer coefficients for which there exist positive integers $a$ and $m$ such that for all positive integers $n\equiv a\pmod m$ , $P(n)$ is nonzero and $$2022\cdot\frac{(n+1)^{n+1} - n^n}{P(n)}$$ is an integer.

Jaedon Whyte, Luke Robitaille, and Pitchayut Saengrungkongka