Let
![x_1](/media/m/9/2/a/92aefd356eeab9982f45f21fb206a2ef.png)
,
![x_2](/media/m/a/a/1/aa16f4edacb7b534405242617406658f.png)
,
![\ldots](/media/m/5/8/5/58542f3cc6046ef3889f8320b7487d60.png)
,
![x_n](/media/m/3/c/5/3c57e4750d576aafa08c9ec1a939cfce.png)
be real numbers satisfying the conditions:
![|x_1 + x_2 + \dots + x_n| = 1](/media/m/0/6/7/0676016e2a87255e97a1eec7d7726a9b.png)
and
![|x_i| \leq \frac{n+1}{2}](/media/m/2/d/3/2d305a87758ca3617fe35602d6001efe.png)
, for
![i = 1, 2, \dots, n](/media/m/2/0/2/2025b72edf49c7ad1282b6131ca72f88.png)
Show that there exists a permutation
![y_1](/media/m/9/2/e/92e328204886ca272c5a4e87b6b32005.png)
,
![y_2](/media/m/a/b/c/abca78e1eef61943a7b852806012be62.png)
,
![\ldots](/media/m/5/8/5/58542f3cc6046ef3889f8320b7487d60.png)
,
![y_n](/media/m/2/a/0/2a08e21c3014a9761a124842ae4a687f.png)
of
![x_1](/media/m/9/2/a/92aefd356eeab9982f45f21fb206a2ef.png)
,
![x_2](/media/m/a/a/1/aa16f4edacb7b534405242617406658f.png)
,
![\ldots](/media/m/5/8/5/58542f3cc6046ef3889f8320b7487d60.png)
,
![x_n](/media/m/3/c/5/3c57e4750d576aafa08c9ec1a939cfce.png)
such that
%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
$$| y_1 + 2 y_2 + \cdots + n y_n | \leq \frac {n + 1}{2}.$$