<s>[INST] Prove the inequality ${1 \over a^3 (b + c)} + {1 \over b^3 (a+c)} + {1 \over c^3 (a+b)} \ge \frac 32 $ given that $abc = 1$ Prove the inequality $${1 \over a^3 (b + c)} + {1 \over b^3 (a+c)} + {1 \over c^3 (a+b)} \ge \frac 32 $$ So, I know a proof for this, but I basically memorized it without understanding. It's
$$ {(\frac 1a + \frac 1b +  \frac 1c)}^2 + {(a(b+c) + b(a+c) + c(a+b))}^-1 = {ab + bc + ac \over 2} \ge {\frac 32} $$
I don't know what inequalities were applied here or how. Another proof would also be appreciated.
 [/INST] we have $$\frac{1}{a^3(b+c)}=\frac{bc}{a^2(b+c)}=\frac{(bc)^2}{a(b+c)}$$ then we get
$$\frac{(bc)^2}{a(b+c)}+\frac{(ac)^2}{b(a+c)}+\frac{(ab)^2}{c(a+b)}\geq \frac{(ab+bc+ac)^2}{2(ab+bc+ac)}\geq \frac{3}{2}$$ by $AM-GM$
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