JBMO 2015 Problem 2

Let $a, b, c$ be positive real numbers such that $a+b+c=3$. Find the minimum value of the expression

$$A=\frac{2-a^{3}}{a}+\frac{2-b^{3}}{b}+\frac{2-c^{3}}{c}$$

$19^{\text {th }}$ Junior Balkan Mathematical Olympiad June 24-29, 2015, Belgrade, Serbia