MathLabs

Problem 4

Find all positive real solutions (x1,x2,x3,x4,x5)(x_1,x_2,x_3,x_4,x_5) of (x12−x3x5)(x22−x3x5)≤0(x_1^2-x_3x_5)(x_2^2-x_3x_5)\le0, (x22−x4x1)(x32−x4x1)≤0(x_2^2-x_4x_1)(x_3^2-x_4x_1)\le0, (x32−x5x2)(x42−x5x2)≤0(x_3^2-x_5x_2)(x_4^2-x_5x_2)\le0, (x42−x1x3)(x52−x1x3)≤0(x_4^2-x_1x_3)(x_5^2-x_1x_3)\le0, and (x52−x2x4)(x12−x2x4)≤0(x_5^2-x_2x_4)(x_1^2-x_2x_4)\le0.
Step 4 of 4: State and verify the solutions
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In plain words

The sum-of-squares identity proves necessity, and substitution proves sufficiency.

(x1,x2,x3,x4,x5)=(t,t,t,t,t),t>0(x_1,x_2,x_3,x_4,x_5)=(t,t,t,t,t),\quad t>0
Detailed analysis

The equalities chain all five variables together. Conversely, every t>0t>0 makes each original product zero, so the complete solution set is (t,t,t,t,t)(t,t,t,t,t).