<s>[INST] Prove that $(\frac{bc+ac+ab}{a+b+c})^{a+b+c} \geq \sqrt{(bc)^a(ac)^b(ab)^c}$ 
Prove that $$\left( \frac{bc+ac+ab}{a+b+c} \right)^{a+b+c} \ge \sqrt{(bc)^a(ac)^b(ab)^c}$$
  where $a,b,c>0$

My attempt:

I couldn't proceed after that.  Please help me in this regard, thanks!
 [/INST] We need to prove that 
$$(a+b+c)\ln\frac{ab+ac+bc}{a+b+c}\geq\frac{1}{2}\sum\limits_{cyc}a\ln{bc}$$ or
$$\ln\left(\frac{ab+ac+bc}{a+b+c}\right)^2\geq\sum\limits_{cyc}\frac{a}{a+b+c}\ln{bc}.$$
But by Jensen $$\sum\limits_{cyc}\frac{a}{a+b+c}\ln{bc}\leq\ln\sum\limits_{cyc}\frac{abc}{a+b+c}$$
Thus, it remains to prove that $$\frac{3abc}{a+b+c}\leq\left(\frac{ab+ac+bc}{a+b+c}\right)^2,$$
which is 
$$\sum\limits_{cyc}c^2(a-b)^2\geq0$$
Done!
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