Source: IMO 1972, Day 2, Problem 4
Tags: algebra, inequality system, IMO, IMO 1972
Find all positive real solutions to: \begin{eqnarray*} (x_1^2-x_3x_5)(x_2^2-x_3x_5) &\le& 0 \\ (x_2^2-x_4x_1)(x_3^2-x_4x_1) &\le& 0 \\ (x_3^2-x_5x_2)(x_4^2-x_5x_2) &\le& 0 \\ (x_4^2-x_1x_3)(x_5^2-x_1x_3) &\le & 0 \\ (x_5^2-x_2x_4)(x_1^2-x_2x_4) &\le& 0 \\ \end{eqnarray*}
sum the inequalities. we have:
$(x_1x_2 - x_1x_4)^2 + (x_2x_3 - x_2x_5)^2 + ... + (x_5x_1 - x_5x_3)^2$
$+ (x_1x_3 - x_1x_5)^2 + ... + (x_5x_2 - x_5x_4)^2 \leq 0$.
so, the squares are all $0$.
the rest is easy. ($x_1 = x_2 = x_3 = x_4 = x_5$)
E.L
Summing we see that:
$$(x_1x_2 - x_1x_4)^2 + (x_2x_3 - x_2x_5)^2 + ... + (x_5x_2 - x_5x_4)^2 \leq 0$$from which we have that:
$$x_1 = x_2 = x_3 = x_4 = x_5$$
Sum and expand to get to a sum of "squares (made of difference of products of $x_i$)" $\leq 0.$ So every "square" is a zero and every $x_i$ is equal.