Question 1
Let $a_{ij}$, $i=1,2,3$; $j=1,2,3$ be real numbers such that $a_{ij}$ is positive for $i=j$ and negative for $i\neq j$.
Prove that there exist positive real numbers $c_{1}, c_{2}, c_{3}$ such that the numbers
$$ a_{11}c_{1}+a_{12}c_{2}+a_{13}c_{3},\qquad a_{21}c_{1}+a_{22}c_{2}+a_{23}c_{3},\qquad a_{31}c_{1}+a_{32}c_{2}+a_{33}c_{3} $$
are all negative, all positive, or all zero.
Prove that there exist positive real numbers $c_{1}, c_{2}, c_{3}$ such that the numbers
$$ a_{11}c_{1}+a_{12}c_{2}+a_{13}c_{3},\qquad a_{21}c_{1}+a_{22}c_{2}+a_{23}c_{3},\qquad a_{31}c_{1}+a_{32}c_{2}+a_{33}c_{3} $$
are all negative, all positive, or all zero.