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Root inequality

Let x, y, z>0 such that \sqrt{xy}+\sqrt{yz}+\sqrt{yz}=1. Prove that

    \[\frac{x^{2}}{x+y}+\frac{y^{2}}{y+z}+\frac{z^{2}}{z+x}\geq \frac{1}{2}\]

Solution

It follows from Cauchy – Schwartz that

    \begin{align*} \frac{x^{2}}{x+y}+\frac{y^{2}}{y+z}+\frac{z^{2}}{z+x} &\geq \frac{(x+y+z)^2}{2(x+y+z)}\\ &=\frac{x+y+z}{2}\\ &\geq \frac{\sqrt{xy}+\sqrt{yz}+\sqrt{yz}}{2}\\ &= \frac{1}{2} \end{align*}

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