Question 170
It is given that real numbers $x_1, x_2, \dots, x_n$ ($n \gt 2$) satisfy $\left|\sum_{i=1}^n x_i\right| \gt 1$, $|x_i| \le 1$ ($i=1, 2, \dots, n$). Prove that there exists a positive integer $k$ such that $\left|\sum_{i=1}^k x_i - \sum_{i=k+1}^n x_i\right| \le 1$. (posed by Leng Gangsong)