MathDB

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 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},\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 either all negative, all positive, or all zero.
Proposed by Kiran Kedlaya, USA

Problem Statement