Školjka

%V0 Let $x_1$, $x_2$, $\ldots$, $x_n$ be real numbers satisfying the conditions: $ |x_1 + x_2 + \dots + x_n| = 1 $ and $|x_i| \leq \frac{n+1}{2}$, for $i = 1, 2, \dots, n$ Show that there exists a permutation $y_1$, $y_2$, $\ldots$, $y_n$ of $x_1$, $x_2$, $\ldots$, $x_n$ such that $$| y_1 + 2 y_2 + \cdots + n y_n | \leq \frac {n + 1}{2}.$$