Let $x_1< x_2< ...< x_n$ be $n$ real numbers such that $\sum\limits_{j=1}^n x_j \neq 0 $.
Do there always exist $n$ integers $a_1,a_2,...,a_n$ such that $\sum\limits_{j=1}^n a_j.x_j <0 $ and $\sum\limits_{j=1}^n a_j.x_{\pi{(j)}} >0 $ for all permutation $\pi$ of $\{1,2,...,n\}$ different from identity?