Question 36
For positive real numbers $a$, $b$ and $c$, prove that
$$
\frac{a^{3}}{a^{2}+a b+b^{2}}+\frac{b^{3}}{b^{2}+b c+c^{2}}+\frac{c^{3}}{c^{2}+c a+a^{2}} \geq \frac{a+b+c}{3}
$$
$$
\frac{a^{3}}{a^{2}+a b+b^{2}}+\frac{b^{3}}{b^{2}+b c+c^{2}}+\frac{c^{3}}{c^{2}+c a+a^{2}} \geq \frac{a+b+c}{3}
$$