ISL 1988 Problem 60

Given integers $a_1, \ldots, a_{10},$ prove that there exist a non-zero sequence $\{x_1, \ldots, x_{10}\}$ such that all $x_i$ belong to $\{-1,0,1\}$ and the number $\sum^{10}_{i=1} x_i \cdot a_i$ is divisible by 1001.