MEMO 2016 ekipno problem 8
We consider the equation $a^2 + b^2 + c^2 + n = abc$, where $a$, $b$, $c$ are positive integers.
\\\\(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 $n = 2016$.
Source: Srednjoeuropska matematička olimpijada 2016, ekipno natjecanje, problem 8