IMO 1997 Problem 3

Let $x_1$ , $x_2$ , $\ldots$ , $x_n$ be real numbers satisfying the conditions: $$\left\{\begin{array}{cccc} |x_1 + x_2 + \cdots + x_n | & = & 1 & \ |x_i| & \leq & \frac {n + 1}{2} & \ \textrm{ for }i = 1, 2, \ldots , n. \end{array} \right.$$ Show that there exists a permutation $y_1$ , $y_2$ , $\ldots$ , $y_n$ of $x_1$ , $x_2$ , $\ldots$ , $x_n$ such that $$| y_1 + 2 y_2 + \cdots + n y_n | \leq \frac {n + 1}{2}.$$