Middle European Mathematical Olympiad 2018 Problem T-8

An integer $n$ is called Silesian if there exist positive integers $a$, $b$ and $c$ such that $$n = \frac{a^2 + b^2 + c^2}{ab + bc + ca}.$$

(a) Prove that there are infinitely many Silesian integers.

(b) Prove that not every positive integer is Silesian.