RO-TST 1996 Problem 2

Find the greatest positive integer $n$ for which there exist $n$ nonnegative integers $x_1, x_2,\ldots , x_n$ , not all zero, such that for any $\varepsilon_1, \varepsilon_2, \ldots, \varepsilon_n$ from the set $\{-1, 0, 1\}$ , not all zero, $\varepsilon_1 x_1 + \varepsilon_2 x_2 + \cdots + \varepsilon_n x_n$ is not divisible by $n^3$ .