ISL 1991 Problem 13

Given any integer $n \geq 2,$ assume that the integers $a_1, a_2, \ldots, a_n$ are not divisible by $n$ and, moreover, that $n$ does not divide $\sum^n_{i=1} a_i.$ Prove that there exist at least $n$ different sequences $(e_1, e_2, \ldots, e_n)$ consisting of zeros or ones such $\sum^n_{i=1} e_i \cdot a_i$ is divisible by $n.$