Question 13
(POL 4) Given any integer $n\ge2$, assume that the integers $a_1,a_2,\ldots,a_n$ are not divisible by $n$ and, moreover, that $n$ does not divide $a_1+a_2+\cdots+a_n$. Prove that there exist at least $n$ different sequences $(e_1,e_2,\ldots,e_n)$ consisting of zeros or ones such that $e_1a_1+e_2a_2+\cdots+e_na_n$ is divisible by $n$.