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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.

Slični zadaci

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