IN-TST 2003 Problem 2

Find all triples $(a,b,c)$ of positive integers such that (i) $a \leq b \leq c$ ; (ii) $\text{gcd}(a,b,c)=1$ ; and (iii) $a^3+b^3+c^3$ is divisible by each of the numbers $a^2b, b^2c, c^2a$ .