<s>[INST] Prove this inequality $\sum\limits_\text{cyc}\sqrt{1-xy}\ge 2$ Let $x,y,z\ge 0$, and $x+y+z=2$, show that
$$\sqrt{1-xy}+\sqrt{1-yz}+\sqrt{1-xz}\ge 2.$$
Mt try: $$\Longleftrightarrow 3-(xy+yz+zx)+2\sum_\text{cyc}\sqrt{(1-xy)(1-yz)}\ge 4$$
or
$$\sum_\text{cyc}\sqrt{(1-xy)(1-yz)}\ge\dfrac{1}{2}(1+xy+yz+zx)$$
 [/INST] Lets assume $z=\min\{x,y,z\}$ then note $\sqrt{1-xy}+\sqrt{1-yz}+\sqrt{1-xz}\geq\sqrt{1-\frac{(2-z)^2}{4}}+2\sqrt[4]{(1-yz)(1-xz)}=\sqrt{z-\frac{z^2}{4}}+2\sqrt[4]{1-z(2-z)+z^2xy}\geq\sqrt{z-\frac{z^2}{4}}+2\sqrt[4]{(1-z)^2+z^4}\geq 2$
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