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 7 of 8: Sum the cyclic estimates
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∑cyc(x+yz)(y+zx)≥∑cycxy+xyz(x+y+z).\sum_{cyc}\sqrt{(x+yz)(y+zx)}\ge\sum_{cyc}\sqrt{xy}+\sqrt{xyz}(\sqrt{x}+\sqrt{y}+\sqrt{z}).
Detailed analysis

Use z square root times xy square root equals xyz square root times z square root, and cycle through x,y,z.