International Mathematical Olympiad 1992 Problem 6

For each positive integer $n$, $S(n)$ is defined to be the greatest integer such that, for every positive integer $k \leq S(n)$, $n^2$ can be written as the sum of $k$ positive squares.

(a) Prove that $S(n) \leq n^2 - 14$ for each $n \geq 4$.

(b) Find an integer $n$ such that $S(n) = n^2 - 14$.

(c) Prove that there are infinitely many integers $n$ such that $S(n) = n^2 - 14$.