Question 45
Let $n \ge 2$ be a positive integer and $x_1, x_2, \dots, x_n$ real numbers ($x_i \ne 0$, for $i=1, 2, \dots, n$) such that $x_1 + x_2 + \dots + x_n = 0$. Prove that there exist distinct positive integers $i$ and $j$ ($i, j \le n$) such that
$$
\frac{1}{2} \le \left| \frac{x_i}{x_j} \right| \le 2.
$$
$$
\frac{1}{2} \le \left| \frac{x_i}{x_j} \right| \le 2.
$$