MathNet 2016-part-02 — 184 problems (EN–VI) Question 39

Let $a$, $b$ and $c$ be positive real numbers satisfying $abc = 1$. Determine the smallest possible value of $\frac{a^3+8}{a^3(b+c)} + \frac{b^3+8}{b^3(c+a)} + \frac{c^3+8}{c^3(a+b)}$.