Question 9
(NET 2) IMO 4. Find all solutions in positive real numbers $x_i$ $(i=1,2,3,4,5)$ of the following system of inequalities:
$$\begin{aligned}(x_1^2-x_3x_5)(x_2^2-x_3x_5)&\le0&&\text{(i)}\\ (x_2^2-x_4x_1)(x_3^2-x_4x_1)&\le0&&\text{(ii)}\\ (x_3^2-x_5x_2)(x_4^2-x_5x_2)&\le0&&\text{(iii)}\\ (x_4^2-x_1x_3)(x_5^2-x_1x_3)&\le0&&\text{(iv)}\\ (x_5^2-x_2x_4)(x_1^2-x_2x_4)&\le0&&\text{(v)}.\end{aligned}$$
$$\begin{aligned}(x_1^2-x_3x_5)(x_2^2-x_3x_5)&\le0&&\text{(i)}\\ (x_2^2-x_4x_1)(x_3^2-x_4x_1)&\le0&&\text{(ii)}\\ (x_3^2-x_5x_2)(x_4^2-x_5x_2)&\le0&&\text{(iii)}\\ (x_4^2-x_1x_3)(x_5^2-x_1x_3)&\le0&&\text{(iv)}\\ (x_5^2-x_2x_4)(x_1^2-x_2x_4)&\le0&&\text{(v)}.\end{aligned}$$