IMO Shortlist 1965 problem 2


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April 2, 2012
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Consider the sytem 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 ofthe coefficients is positive.

Prove that the given system has only the solution x_1=x_2=x_3=0.
Source: Međunarodna matematička olimpijada, shortlist 1965