Question 40
Determine all positive integers $n \ge 2$ which satisfy the following condition: For all integers $a_1, a_2, \dots, a_n$ such that their sum is not divisible by $n$, there exists an integer $i \in \{1, 2, \dots, n\}$ such that none of the $n$ numbers
$$
a_i,\ a_i + a_{i+1},\ \dots,\ a_i + a_{i+1} + \dots + a_{i+n-1}
$$
is divisible by $n$, where we define $a_i = a_{i-n}$ for the case $i \gt n$.
$$
a_i,\ a_i + a_{i+1},\ \dots,\ a_i + a_{i+1} + \dots + a_{i+n-1}
$$
is divisible by $n$, where we define $a_i = a_{i-n}$ for the case $i \gt n$.