2003 IMO 第 1 题

A1 (USA) Let $a_{i j}, i=1,2,3, j=1,2,3$, be real numbers such that $a_{i j}$ 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 $$a_{11} c_{1}+a_{12} c_{2}+a_{13} c_{3}, \quad a_{21} c_{1}+a_{22} c_{2}+a_{23} c_{3}, \quad a_{31} c_{1}+a_{32} c_{2}+a_{33} c_{3}$$ are all negative, all positive, or all zero.

A1 (美国) 设 $a_{i j}, i=1,2,3, j=1,2,3$ 为实数，使得 $a_{i j}$ 对于 $i=j$ 为正，对于 $i \neq j$ 为负。证明存在正实数 $c_{1}, c_{2}, c_{3}$ 使得数 $$a_{11} c_{1}+a_{12} c_{2}+a_{13} c_{3}, \quad a_{21} c_{1}+a_{22} c_{2}+a_{23} c_{3}, \quad a_{31} c_{1}+a_{32} c_{2}+a_{33} c_{3}$$ 全部为负、全部为正或全部为零。