<s>[INST] Prove $\frac{a+b}{\sqrt{c}}+\frac{a+c}{\sqrt{b}}+ \frac{b+c}{\sqrt{a}} \geq 2(\sqrt{a} + \sqrt{b} +\sqrt{c})$ Could anyone advise me on how to prove this inequality:
$$\dfrac{a+b}{\sqrt{c}}+\dfrac{a+c}{\sqrt{b}}+ \dfrac{b+c}{\sqrt{a}} \geq 2(\sqrt{a} + \sqrt{b} +\sqrt{c}),$$
where  $a,b,c $ are any positive real numbers.
Do I use the AM-GM inequality somewhere?
Thank you.
 [/INST] From AM-GM you have 
$$ \frac{a}{\sqrt{b}} +\frac{a}{\sqrt{b}}  +\frac{b}{\sqrt{a}}   \geq 3 \sqrt{a}     $$
and
$$ \frac{a}{\sqrt{c}} +\frac{a}{\sqrt{c}}  +\frac{c}{\sqrt{a}}   \geq 3 \sqrt{a}     $$
Doing this for the other variables and summing respectively you get
$$3\left (  \dfrac{a+b}{\sqrt{c}}+\dfrac{a+c}{\sqrt{b}}+ \dfrac{b+c}{\sqrt{a}}  \right)       \geq 6(  \sqrt{a} + \sqrt{b} +\sqrt{c}   )$$
Simplifying gives the desired inequality.
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