<s>[INST] Prove the inequality $\sum\limits_{\text{cyc}} \frac{x}{y+z+\sqrt{x}}≥ \frac{(x+y+z)^2}{2(xy+yz+zx)+x\sqrt{x}+y\sqrt{y}+z\sqrt{z}}$ How do I show the following?
$$
\frac{x}{y+z+\sqrt{x}}+\frac{y}{z+x+\sqrt{y}}+\frac{z}{x+y+\sqrt{z}}≥ \frac{(x+y+z)^2}{2(xy+yz+zx)+x\sqrt{x}+y\sqrt{y}+z\sqrt{z}}
$$
Any hint would do; I don't require a complete solution.
 [/INST] use that e.g. for $x>0$ we have $\frac{x}{y+z+\sqrt{x}}=\frac{x^2}{xy+xz+x\sqrt{x}}$ and use Titu's Lemma$
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