2021 IMO Shortlist N6

Determine all integers $n \geq 2$ with the following property: every $n$ pairwise distinct integers whose sum is not divisible by $n$ can be arranged in some order $a_{1}, a_{2}, \ldots, a_{n}$ so that $n$ divides $1 \cdot a_{1}+2 \cdot a_{2}+\cdots+n \cdot a_{n}$.

使用以下属性确定所有整数 $n \geq 2$：每个 $n$ 两对不同的整数，其总和不能被 $n$ 整除，可以按某种顺序排列 $a_{1}, a_{2}, \ldots, a_{n}$，以便 $n$ 除以 $1 \cdot a_{1}+2 \cdot a_{2}+\cdots+n \cdot a_{n}$。