OIM 2019 Problem 6

Let $a_1, a_2, \dots, a_{2019}$ be positive integers and $P$ a polynomial with integer coefficients such that, for every positive integer $n$ , $$P(n) \text{ divides } a_1^n+a_2^n+\dots+a_{2019}^n.$$ Prove that $P$ is a constant polynomial.