Problem

Source: ELMO 2022 P2

Tags: algebra, polynomial, ELMO 2022



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