2024 IMO Shortlist N6

Let $n$ be a positive integer. We say that a polynomial $P$ with integer coefficients is $n$-good if there exists a polynomial $Q$ of degree $2$ with integer coefficients such that $Q(k)(P(k)+Q(k))$ is never divisible by $n$ for any integer $k$.

Determine all integers $n$ such that every polynomial with integer coefficients is an $n$-good polynomial.

设 $n$ 为正整数。若对一个整数系数多项式 $P$，存在一个二次整数系数多项式 $Q$，使得对任意整数 $k$，$Q(k)(P(k)+Q(k))$ 都不被 $n$ 整除，则称 $P$ 是 $n$-good 的。

确定所有整数 $n$，使得每个整数系数多项式都是 $n$-good 多项式。