<s>[INST] Proving a small inequality I am given that $a^3+b^3+c^3=3$ where a, b, c are positive numbers and I need to prove that
$$\frac {3(ab+bc+ac)+a^3c^2+b^3a^2+c^3b^2}{(a+b)(b+c)(a+c)}\ge \frac {3}{2}$$
At first it seems to me that the inequality might be wrong. I have tried using the Cauchy Schwarz , AM GM and some algebraic manipulations to reach the inequality but none of them helped me out. 
Can somebody around  here help me to prove this small inequality. Thanks in advance.
 [/INST] We need to prove that
$$\sum_{cyc}\frac{3ab+a^3c^2}{\prod\limits_{cyc}(a+b)}\geq\frac{3}{2}$$ or
$$\sum_{cyc}\frac{(a^3+b^3+c^3)ab+a^3c^2}{\prod\limits_{cyc}(a+b)}\geq\frac{3}{2}$$ or
$$\sum_{cyc}\frac{a^4b+a^4c+a^3bc+a^3c^2}{\prod\limits_{cyc}(a+b)}\geq\frac{3}{2}$$ or
$$\sum_{cyc}\frac{a^3(ab+ac+bc+c^2)}{\prod\limits_{cyc}(a+b)}\geq\frac{3}{2}$$ or
$$\sum_{cyc}\frac{a^3(b+c)(c+a)}{\prod\limits_{cyc}(a+b)}\geq\frac{3}{2}$$or
$$\sum_{cyc}\frac{a^3}{a+b}\geq\frac{3}{2}$$ and see here:
If $a^3+b^3+c^3=3$ so $\frac{a^3}{a+b}+\frac{b^3}{b+c}+\frac{c^3}{c+a}\geq\frac{3}{2}$
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