International Mathematical Olympiad 1965 Problem 2

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 $x_1 = x_2 = x_3 = 0$.