<s>[INST] Inequality. $\frac{a^3}{a+b}+\frac{b^3}{b+c}+\frac{c^3}{c+a} \geq \frac{ab+bc+ca}{2}$ prove the following inequality: 
$$\frac{a^3}{a+b}+\frac{b^3}{b+c}+\frac{c^3}{c+a} \geq \frac{ab+bc+ca}{2},$$ for $a,b,c$ real positive numbers. 
Thanks :) 
 [/INST] Note that using the AM-GM inequality, we have
$$\frac{a^3}{a+b}=a^2-\frac{a^2b}{a+b}\geq a^2-\frac{a^2b}{2\sqrt{ab}}=a^2-\frac 12\sqrt{a^3b},$$
And so summing up cyclically, it is enough to check that
$$2(a^2+b^2+c^2)\geq\sqrt{a^3b}+\sqrt{b^3c}+\sqrt{c^3a}+ab+bc+ca,$$
Which is obviously true since $\sqrt{a^3b}\leq \dfrac{a^2+ab}2$ and $ab+bc+ca\leq a^2+b^2+c^2$ from the AM-GM inequality.
Equality holds iff $a=b=c.$
$\Box$
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