Problem

Source: Ukrainian Mathematical Olympiad 2024. Day 2, Problem 11.8

Tags: number theory, polynomial



Find all polynomials $P(x)$ with integer coefficients, such that for each of them there exists a positive integer $N$, such that for any positive integer $n\geq N$, number $P(n)$ is a positive integer and a divisor of $n!$. Proposed by Mykyta Kharin