IMO 1965/A2 solution

IMO 1965

Problem A2

The coefficients a_ij of the following equations

a₁₁x₁ + a₁₂ x₂+ a₁₃ x₃ = 0
a₂₁x₁ + a₂₂x₂ + a₂₃x₃ = 0
a₃₁x₁ + a₃₂x₂ + a₃₃x₃ = 0

satisfy the following: (a) a₁₁, a₂₂, a₃₃ are positive, (b) other a_ij are negative, (c) the sum of the coefficients in each equation is positive. Prove that the only solution is x₁ = x₂ = x₃ = 0.

Solution

The slog solution is to multiply out the determinant and show it is non-zero. A slicker solution is to take the x_i with the largest absolute value. Say |x₁| ≥ |x₂|, |x₃|. Then looking at the first equation we have an immediate contradiction, since the first term has larger absolute value than the sum of the absolute values of the second two terms.

Solutions are also available in: Samuel L Greitzer, International Mathematical Olympiads 1959-1977, MAA 1978, and in István Reiman, International Mathematical Olympiad 1959-1999, ISBN 189-8855-48-X.

7th IMO 1965