<s>[INST] How to prove an inequality I didn't know where can I start because it doesn't fit to any theories or formulas.

If $a>0$, $b>0$, $c>0$ and $a+b+c=1$, prove that:
$$\frac{a^3}{a^2+b^2}+\frac{b^3}{b^2+c^2}+\frac{c^3}{c^2+a^2}\geq \frac{1}{2}$$

 [/INST] We need to prove that
$$\sum_{cyc}\frac{a^3}{a^2+b^2}\geq\frac{1}{2}$$ or
$$\sum_{cyc}\frac{a^3}{a^2+b^2}\geq\frac{a+b+c}{2}$$ or
$$\sum_{cyc}\left(\frac{a^3}{a^2+b^2}-\frac{a}{2}\right)\geq0$$ or
$$\sum_{cyc}\frac{a(a^2-b^2)}{a^2+b^2}\geq0$$ or
$$\sum_{cyc}\left(\frac{a(a^2-b^2)}{a^2+b^2}-(a-b)\right)\geq0$$ or
$$\sum_{cyc}\frac{b(a-b)^2}{a^2+b^2}\geq0.$$
Done!
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