Question 163
Let $a$, $b$, $c$, $d$ be non-negative real numbers satisfying $a + b + c + d = 3$. Prove that
$$
\frac{a}{1 + 2b^3} + \frac{b}{1 + 2c^3} + \frac{c}{1 + 2d^3} + \frac{d}{1 + 2a^3} \ge \frac{a^2 + b^2 + c^2 + d^2}{3}.
$$
When does the equality hold?
$$
\frac{a}{1 + 2b^3} + \frac{b}{1 + 2c^3} + \frac{c}{1 + 2d^3} + \frac{d}{1 + 2a^3} \ge \frac{a^2 + b^2 + c^2 + d^2}{3}.
$$
When does the equality hold?