MathLabs

Problem 4

Let x,y,zx,y,z be positive numbers such that 1x+1y+1z=1\frac1x+\frac1y+\frac1z=1. Show that x+yz+y+zx+z+xy≥xyz+x+y+z\sqrt{x+yz}+\sqrt{y+zx}+\sqrt{z+xy}\ge\sqrt{xyz}+\sqrt{x}+\sqrt{y}+\sqrt{z}.
Step 2 of 8: Name the two sides
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L=∑cycx+yz,R=xyz+x+y+z.L=\sum_{cyc}\sqrt{x+yz},\qquad R=\sqrt{xyz}+\sqrt{x}+\sqrt{y}+\sqrt{z}.
Detailed analysis

It is enough to prove L is at least R; both are positive, so we may compare their squares.