International Mathematical Olympiad 1972 Problem 4

AlgebraInequalitiesSystems

Find all solutions (x1,x2,x3,x4,x5)(x_1, x_2, x_3, x_4, x_5) of the system of inequalities (x12−x3x5)(x22−x3x5)≤0(x_1^2 - x_3x_5)(x_2^2 - x_3x_5) \leq 0 (x22−x4x1)(x32−x4x1)≤0(x_2^2 - x_4x_1)(x_3^2 - x_4x_1) \leq 0 (x32−x5x2)(x42−x5x2)≤0(x_3^2 - x_5x_2)(x_4^2 - x_5x_2) \leq 0 (x42−x1x3)(x52−x1x3)≤0(x_4^2 - x_1x_3)(x_5^2 - x_1x_3) \leq 0 (x52−x2x4)(x12−x2x4)≤0(x_5^2 - x_2x_4)(x_1^2 - x_2x_4) \leq 0 where x1,x2,x3,x4,x5x_1, x_2, x_3, x_4, x_5 are positive real numbers.