<s>[INST] What is the solution to $\frac1{a^2 +2} + \frac1{b^2 +2} + \frac1{c^2 +2} \le \frac{\sqrt2}{2}\frac{\sqrt a+\sqrt b+\sqrt c}{\sqrt{abc}}$ one of my friends asked me if I could solve him a mathematics problem.
It looks like this:
$$\frac1{a^2 +2} + \frac1{b^2 +2} + \frac1{c^2 +2} \le \frac{\sqrt2}{2}\frac{\sqrt a+\sqrt b+\sqrt c}{\sqrt{abc}}$$
As I think it looks like an inequality between means, hope it helps.
And sorry for my bad english by the way. :)
 [/INST] Without loss of generality, we assume that $0 < a \leq b \leq c.$ Then, by AM-GM we have, 
$$
\frac{a^2 + (\sqrt{2})^2}{2} \geq a\sqrt{2}, 
$$
and so 
$$
\frac{1}{a^2 + 2} + \frac{1}{b^2+2} + \frac{1}{c^2+2} \leq \frac{1}{2\sqrt{2}}\left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right).
$$
Now, 
$
\frac{1}{\sqrt{c}} \leq \frac{1}{\sqrt{b}} \leq \frac{1}{\sqrt{a}},
$
so using the rearrangement inequalities, and adding, we obtain the desired result. 
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