Problem 4

Find all positive real solutions

(x_1,x_2,x_3,x_4,x_5)

(x_1^2-x_3x_5)(x_2^2-x_3x_5)\le0

(x_2^2-x_4x_1)(x_3^2-x_4x_1)\le0

(x_3^2-x_5x_2)(x_4^2-x_5x_2)\le0

(x_4^2-x_1x_3)(x_5^2-x_1x_3)\le0

, and

(x_5^2-x_2x_4)(x_1^2-x_2x_4)\le0

Step 3 of 4: Cancel positive factors

In plain words

Positivity lets product equalities become variable equalities.

x_1x_2=x_1x_4,\quad x_2x_3=x_2x_5,\quad x_3x_4=x_3x_1,\quad x_4x_5=x_4x_2

Detailed analysis

All ten squares must vanish. Since every $x_i>0$ , the displayed equalities imply $x_2=x_4$ , $x_3=x_5$ , $x_4=x_1$ , and $x_5=x_2$ .