<s>[INST] Given three positive numbers $a,b,c$. Prove that $\sum\limits_{sym}\frac{a+b}{c}\geqq 2\sqrt{(\sum\limits_{sym}a)(\sum\limits_{sym}\frac{a}{bc}})$ . 
(A problem due to Mr. Le Khanh Sy). Given three positive numbers $a, b, c$. Prove that
  $$\sum\limits_{sym}\frac{a+ b}{c}\geqq 2\sqrt{(\sum\limits_{sym}a)(\sum\limits_{sym}\frac{a}{bc}})$$

I'm eagerly interested in learning one method which assumes $c\not\equiv {\rm mid}(\!a, b, c\!)$. But if $c\equiv {\rm mid}(\!a, b, c\!)$:
$$2\sqrt{(\!\frac{a}{bc}+ \frac{b}{ca}+ \frac{c}{ab}\!)(\!a+ b+ c\!)}\leqq c(\!\frac{a}{bc}+ \frac{b}{ca}+ \frac{c}{ab}\!)+ \frac{a+ b+ c}{c}= \frac{a+ b}{c}+ \frac{a}{b}+ \frac{b}{a}+ 1+ \frac{c^{2}}{ab}$$
We need to prove
$$\begin{align} \frac{a+ b}{c}+ \frac{a}{b}+ \frac{b}{a}+ 1+ \frac{c^{2}}{ab}\leqq \frac{a+ b}{c}+ \frac{b+ c}{a} & + \frac{c+ a}{b}\Leftrightarrow 1+ \frac{c^{2}}{ab}\leqq \frac{c}{a}+ \frac{c}{b}\Leftrightarrow \\ & \Leftrightarrow (\frac{c}{a}- 1)(\frac{c}{b}- 1)\leqq 0\Leftrightarrow \frac{(c- a)(c- b)}{ab}\leqq 0 \end{align}$$
Who can teach me what would we do if $c\not\equiv {\rm mid}(\!a, b, c\!)$ ? I am goin' to set a bounty, thank u so much
 [/INST] By the Vornicu Schur, we have
$$\left( {\frac{a}{b} + \frac{b}{c} + \frac{c}{a}} \right)\left( {\frac{a}{c} + \frac{c}{b} + \frac{b}{a}} \right) - \left( {a + b + c} \right)\left( {\frac{a}{{bc}} + \frac{b}{{ca}} + \frac{c}{{ab}}} \right) = \sum \frac{(a-b)(a-c)}{a^2} \geqslant 0.$$
Theforere we will show that
$$\frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b} \geqslant 2\sqrt{\left( {\frac{a}{b} + \frac{b}{c} + \frac{c}{a}} \right)\left( {\frac{a}{c} + \frac{c}{b} + \frac{b}{a}} \right)},$$
equivalent to
$$\left( {\frac{a}{b} + \frac{b}{c} + \frac{c}{a}} \right)+\left( {\frac{a}{c} + \frac{c}{b} + \frac{b}{a}} \right) \geqslant 2\sqrt{\left( {\frac{a}{b} + \frac{b}{c} + \frac{c}{a}} \right)\left( {\frac{a}{c} + \frac{c}{b} + \frac{b}{a}} \right)}.$$
It's $x+y \geqslant 2\sqrt{xy}.$ The proof is completed.
Note. The inequality follows from the famous inequality
$$(x+y+z)^2 \geqslant 4(yz\sin^{2}A + zx\sin^{2}B + xy\sin^{2}C).$$
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