CWMO 2005 Problem 4

Given is the positive integer $n > 2$ . Real numbers $\mid x_i \mid \leq 1$ ( $i = 1, 2, ..., n$ ) satisfying $\mid \sum_{i=1}^{n}x_i \mid > 1$ . Prove that there exists positive integer $k$ such that $\mid \sum_{i=1}^{k}x_i - \sum_{i=k+1}^{n}x_i \mid \leq 1$ .