MathNet 2010 — Part 04 Question 32

Let $a$, $b$, $c$ be positive real numbers such that $a + b + c = 3$. Prove the inequality
$$
\frac{a^3+2}{b+2} + \frac{b^3+2}{c+2} + \frac{c^3+2}{a+2} \ge 3.
$$