Middle European Mathematical Olympiad 2016 Problem T-8

We consider the equation $a^2 + b^2 + c^2 + n = abc$, where $a, b, c$ are positive integers.

Prove:

(a) There are no solutions $(a,b,c)$ for $n = 2017$.

(b) For $n = 2016$, $a$ must be divisible by $3$ for every solution $(a, b, c)$.

(c) The equation has infinitely many solutions $(a, b, c)$ for $n = 2016$.