2017 IMO Shortlist N3

Determine all integers $n \geq 2$ with the following property: for any integers $a_{1}, a_{2}, \ldots, a_{n}$ whose sum is not divisible by $n$, there exists an index $1 \leq i \leq n$ such that none of the numbers $$a_{i}, a_{i}+a_{i+1}, \ldots, a_{i}+a_{i+1}+\cdots+a_{i+n-1}$$ is divisible by $n$. (We let $a_{i}=a_{i-n}$ when $i>n$.) (Thailand)

确定所有整数 $n \geq 2$ 具有以下属性：对于任何整数 $a_{1}, a_{2}, \ldots, a_{n}$ 其总和不能被 $n$ 整除，存在索引 $1 \leq i \leq n$ 使得数字 $$a_{i}, a_{i}+a_{i+1}, \ldots, a_{i}+a_{i+1}+\cdots+a_{i+n-1}$$ 可以被 $n$ 整除。 (当 $i>n$ 时，我们令 $a_{i}=a_{i-n}$。)(泰国)