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 4 of 8: Expand the target square
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R2=xyz+x+y+z+2xyz(x+y+z)+2(xy+yz+zx).R^2=xyz+x+y+z+2\sqrt{xyz}(\sqrt{x}+\sqrt{y}+\sqrt{z})+2(\sqrt{xy}+\sqrt{yz}+\sqrt{zx}).
Detailed analysis

This is the ordinary expansion of the four positive summands in R.