<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] Need to prove$:$ $$\displaystyle \left( {\dfrac {a+b}{c}}+{\dfrac {b+c}{a}}+{\dfrac {a+c}{b}} \right) ^{2
}\geqslant 4 \left( a+b+c \right)  \left( {\dfrac {a}{bc}}+{\dfrac {b}{ac}}+{
\dfrac {c}{ab}} \right) $$
Or $$\displaystyle \,{\frac { 2\left( ab-2\,ac+bc \right) ^{2}}{ \left( {a}^{2}+{c}^{2} \right) {b}^{2}}}+{\frac {2 \left( a-c \right) ^{2} \left[2\,b{a}^{2}+abc+2\,b{c}^{2}-ac(a+c) \right]^{2}}{ \left( {a}^{2}+{ c}^{2} \right)  \left( {a}^{2}+{b}^{2}+{c}^{2} \right) {c}^{2}{a}^{2}} }+$$
$$+{\dfrac { \left( a-b \right) ^{2} \left( b-c \right) ^{2} \left( a-c
 \right) ^{2} \left( {a}^{2}+4\,ab+4\,ac+{b}^{2}+4\,bc+{c}^{2}
 \right) }{ \left( {a}^{2}+{b}^{2}+{c}^{2} \right) {c}^{2}{a}^{2}{b}^{
2}}}\geqslant 0$$
Which is true.
Text for the SOS's expression: Click here.
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