Question 64
Let $a$, $b$, $c$ be positive real numbers. Prove that
$$
\frac{a^{3}}{a^{2}+b c}+\frac{b^{3}}{b^{2}+c a}+\frac{c^{3}}{c^{2}+a b} \geq \frac{\left(a^{2}+b^{2}+c^{2}\right)(a b+b c+c a)}{a^{3}+b^{3}+c^{3}+3 a b c}
$$
When will the equality hold?
$$
\frac{a^{3}}{a^{2}+b c}+\frac{b^{3}}{b^{2}+c a}+\frac{c^{3}}{c^{2}+a b} \geq \frac{\left(a^{2}+b^{2}+c^{2}\right)(a b+b c+c a)}{a^{3}+b^{3}+c^{3}+3 a b c}
$$
When will the equality hold?