MathLabs

Problem 3

Let x,y,z>0x, y, z > 0 satisfy xyz≥1xyz\ge 1. Prove that x5−x2x5+y2+z2+y5−y2x2+y5+z2+z5−z2x2+y2+z5≥0.\frac{x^5-x^2}{x^5+y^2+z^2} + \frac{y^5-y^2}{x^2+y^5+z^2} + \frac{z^5-z^2}{x^2+y^2+z^5} \ge 0.
Step 4 of 5: It remains to prove a simpler inequality
Previous stepNext step
x2+y2+z2 ≥ 1x+1y+1zx^2+y^2+z^2\ \ge\ \frac1x+\frac1y+\frac1z
Detailed analysis

By the previous step it suffices to prove this inequality; once it holds, the right-hand side of Step 3 is ≥0\ge0, which finishes the problem.