Question 11
Let $a_1, a_2, \dots, a_n$ be positive integers. Assume that for some positive integer $k$, $k \lt n$, the following is true: if we choose any $k$ of the given $n$ numbers their sum is divisible by $n$. Prove that $a_1 + a_2 + \dots + a_n$ is also divisible by $n$.