<s>[INST] How to prove that $a^2(1+b^2)+b^2(1+c^2)+c^2(1+a^2)\geq6abc$ Help me prove $a^2(1+b^2)+b^2(1+c^2)+c^2(1+a^2)\geq6abc$
 [/INST] LHS=$\sum a^2\ +\ \sum(ab)^2$
Applying A.M. ≥ G.M.,
$\sum a^2 ≥ 3(abc)^{\frac{2}{3}} $
$\sum (ab)^2 ≥ 3(abc)^{\frac{4}{3}} $
Taking summation,$ LHS ≥ 3((abc)^{\frac{2}{3}}  + (abc)^{\frac{4}{3}}) $
But $(abc)^{\frac{2}{3}}  + (abc)^{\frac{4}{3}}$ ≥ 2 abc  (applying A.M. ≥ G.M.,)
LHS ≥ 3(2abc)
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