Difference between revisions of "1965 IMO Problems/Problem 2"

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Revision as of 11:43, 16 July 2009

Problem

Consider the system of equations \[a_{11}x_1 + a_{12}x_2 + a_{13}x_3 = 0\] \[a_{21}x_1 + a_{22}x_2 + a_{23}x_3 = 0\] \[a_{31}x_1 + a_{32}x_2 + a_{33}x_3 = 0\] with unknowns $x_1$, $x_2$, $x_3$. The coefficients satisfy the conditions:

(a) $a_{11}$, $a_{22}$, $a_{33}$ are positive numbers;

(b) the remaining coefficients are negative numbers;

(c) in each equation, the sum of the coefficients is positive.

Prove that the given system has only the solution $x_1 = x_2 = x_3 = 0$.

Solution

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