PE-MO 2014 Problem 3

a) Let $a, b, c$ be positive integers such that $ab + b + 1$ , $bc + c + 1$ and $ca + a + 1$ are divisors of the number $abc - 1$ , prove that $a = b = c$ .

b) Find all triples $(a, b, c)$ of positive integers such that the product $$(ab - b + 1)(bc - c + 1)(ca - a + 1)$$ is a divisor of the number $(abc + 1)^2$ .