Thread: Problem 28
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Old June 25th, 2007, 07:06 AM
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For x,y,z\in\mathbf{R} the inequality is not tight.
We have x^2+y^2+z^2\geq xy+xz+yz\Rightarrow
\Rightarrow (x+y+z)^2-2(xy+xz+yz)\geq xy+xz+yz\Rightarrow
\Rightarrow xy+xz+yz\leq \frac{1}{3}<\frac{1}{2}.
The equality stands for x=y=z=\frac{1}{3}.