\sqrt{x}+\sqrt{y}+\sqrt{z}=1

[ghenghea1]

$\sqrt{x}+\sqrt{y}+\sqrt{z}=1 \Rightarrow \frac{{{x}^{2}}+yz}{\sqrt{2{{x}^{2}}(y+z)}}+\frac{{{y}^{2}}+zx}{\sqrt{2{{y}^{2}}(z+x)}}+ \frac{{{z}^{2}}+xy}{\sqrt{2{{z}^{2}}(x+y)}}\geq 1.$