Question 22
Determine all integers $n \geq 2$ with the following property:
For arbitrary, not necessarily distinct integers $m_{1}, m_{2}, \ldots, m_{n}$ whose sum is not divisible by $n$, there exists an index $i$ $(1 \leq i \leq n)$ such that none of the numbers
$$
m_{i},\ m_{i}+m_{i+1},\ m_{i}+m_{i+1}+m_{i+2},\ \ldots,\ m_{i}+m_{i+1}+\ldots+m_{i+n-1}
$$
is divisible by $n$. (Here $m_{i}=m_{i-n}$ for $i\gt n$.)
For arbitrary, not necessarily distinct integers $m_{1}, m_{2}, \ldots, m_{n}$ whose sum is not divisible by $n$, there exists an index $i$ $(1 \leq i \leq n)$ such that none of the numbers
$$
m_{i},\ m_{i}+m_{i+1},\ m_{i}+m_{i+1}+m_{i+2},\ \ldots,\ m_{i}+m_{i+1}+\ldots+m_{i+n-1}
$$
is divisible by $n$. (Here $m_{i}=m_{i-n}$ for $i\gt n$.)