CMO 2002 Problem 3

Prove that for all positive real numbers $a, b$, and $c$,

$$\frac{a^{3}}{b c}+\frac{b^{3}}{c a}+\frac{c^{3}}{a b} \geq a+b+c$$

and determine when equality occurs.

Each of the inequalities used in the solutions below has the property that equality holds if and only if $a=b=c$. Thus equality holds for the given inequality if and only if $a=b=c$.