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 6 of 8: Take square roots
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(x+yz)(y+zx)≥(1+z)xy=xy+xyzz.\sqrt{(x+yz)(y+zx)}\ge(1+z)\sqrt{xy}=\sqrt{xy}+\sqrt{xyz}\sqrt z.
Detailed analysis

All quantities are positive. The cyclic versions give analogous inequalities with x and y in the other positions.