<s>[INST] Proving inequalities (1) 
Let $a, b, c > 0$ $;abc=8$  ,
  $$\frac{a^2}{\sqrt{(1+a^3)(1+b^3)}}
+ \frac{b^2}{\sqrt{(1+b^3)(1+c^3)}}
+ \frac{c^2}{\sqrt{(1+c^3)(1+a^3)}} \ge \frac43
$$
  (ref: original image)

I tried using AM-GM to eliminate the square root on the bottom but I am stuck, what is the general strategy here and maybe I can use its symmetry?
 [/INST] By AM-GM $$1+a^3=(1+a)(1-a+a^2)\leq\left(\frac{1+a+1-a+a^2}{2}\right)^2=\frac{(2+a^2)^2}{4}.$$
Thus, it's enough to prove that
$$\sum_{cyc}\frac{a^2}{(2+a^2)(2+b^2)}\geq\frac{1}{3}$$ or 
$$3\sum_{cyc}a^2(2+c^2)\geq\prod_{cyc}(2+a^2)$$ or
$$\sum_{cyc}(a^2b^2+2a^2)\geq72,$$ which is true by AM-GM:
$$\sum_{cyc}(a^2b^2+2a^2)\geq3\sqrt[3]{a^4b^4c^4}+6\sqrt[3]{a^2b^2c^2}=3\cdot16+6\cdot4=72.$$
Done!
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