IMO 1966/B2 solution

IMO 1966

Problem B2

Solve the equations:

|a_i - a₁| x₁ + |a_i - a₂| x₂ + |a_i - a₃| x₃ + |a_i - a₄| x₄ = 1, i = 1, 2, 3, 4, where a_i are distinct reals.

Answer

x₁ = 1/(a₁ - a₄), x₂ = x₃ = 0, x₄ = 1/(a₁ - a₄).

Solution

Take a₁ > a₂ > a₃ > a₄. Subtracting the equation for i=2 from that for i=1 and dividing by (a₁ - a₂) we get:

- x₁ + x₂ + x₃ + x₄ = 0.

Subtracting the equation for i=4 from that for i=3 and dividing by (a₃ - a₄) we get:

- x₁ - x₂ - x₃ + x₄ = 0.

Hence x₁ = x₄. Subtracting the equation for i=3 from that for i=2 and dividing by (a₂ - a₃) we get:

- x₁ - x₂ + x₃ + x₄ = 0.

Hence x₂ = x₃ = 0, and x₁ = x₄ = 1/(a₁ - a₄).

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.

8th IMO 1966