ISL 2021 N6

Determine all integers $n \geqslant 2$ 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 $1 \cdot a_{1}+2 \cdot a_{2}+\cdots+n \cdot a_{n}$.