<s>[INST] Proof of an inequality 
Let $a$, $b$ and $c$ be positive numbers. Prove that:
  $$\frac{\sqrt{a+b+c}+\sqrt{a}}{b+c} + \frac{\sqrt{a+b+c}+\sqrt{b}}{c+a} + \frac{\sqrt{a+b+c}+\sqrt{c}}{a+b} \geq \frac{9+3\sqrt{3}}{2\sqrt{a+b+c}}$$

 [/INST] As the inequality is homogeneous, we can set $a+b+c = 1$, say.  Then we have to show the cyclic sum:
$$\sum_{cyc} \frac{1+\sqrt{a}}{1-a} = \sum_{cyc} \frac1{1- \sqrt a} \ge \frac{9+3\sqrt3}2$$
To show this, it is sufficient to show that 
$$f(x) = \frac1{1-\sqrt x} - \frac{3+\sqrt 3}2 - k(\tfrac13-x) \ge 0$$
for some $k \in \mathbb R$ and $x \in (0, 1)$, as the inequality is equivalent to $f(a)+f(b)+f(c) \ge 0$.  
We find that $k = \frac34(3+2\sqrt 3)$ works, as then 
$$f(x) = \frac{(3 + 2 \sqrt3) \left(\sqrt3 - 3 \sqrt x \right)^2 (-3 + 2 \sqrt3 + 3 \sqrt x)}{36(1 - \sqrt x)} \ge 0, \quad \forall x \in (0, 1)$$
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