1992 IMO Problems/Problem 6

Problem

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

(a) Prove that $S(n) \le n^{2}-14$ for each $n \ge 4$ .

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

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

Solution

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1992 IMO Problems/Problem 6

Problem

Solution