Školjka

%V0 Let $a_{ij}$ (with the indices $i$ and $j$ from the set $\left\{1,\ 2,\ 3\right\}$) be real numbers such that $a_{ij}>0$ for $i = j$; $a_{ij}<0$ for $i\neq j$. Prove the existence of positive real numbers $c_{1}$, $c_{2}$, $c_{3}$ such that the numbers $a_{11}c_{1}+a_{12}c_{2}+a_{13}c_{3}$, $a_{21}c_{1}+a_{22}c_{2}+a_{23}c_{3}$, $a_{31}c_{1}+a_{32}c_{2}+a_{33}c_{3}$ are either all negative, or all zero, or all positive.