MathNet 2011 Question 58

Let $a$, $b$ and $c$ be coprime positive integers so that $a^2 \mid (b^3 + c^3)$, $b^2 \mid (a^3 + c^3)$ and $c^2 \mid (a^3 + b^3)$. Find the values of $a$, $b$ and $c$. (posed by Yang Xiaoming)