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

Let $c\neq  {\rm mid}\{a,b,c\}$ and $c'={\rm mid}\{a,b,c\}.$
Now, let another variables be $a'$ and $b'$.
Thus, since our inequality does not depend on any permutations of $a'$, $b'$ and $c'$,  we need to prove that:
$$\frac{a'+b'}{c'}+\frac{b'+c'}{a'}+\frac{c'+a'}{b'}\geq2\sqrt{(a'+b'+c')\left(\frac{a'}{b'c'}+\frac{b'}{c'a'}+\frac{c'}{a'b'}\right)},$$ where $a'$, $b'$ and $c'$ are positives such that $c'={\rm mid}\{a',b',c'\},$ which you made already.
