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2023 Romanian Master of Mathematics Shortlist, N1
Let $n$ be a positive integer. Let $S$ be a set of ordered pairs $(x, y)$ such that $1\leq x \leq n$ and $0 \leq y \leq n$ in each pair, and there are no pairs $(a, b)$ and $(c, d)$ of different elements in $S$ such that $a^2+b^2$ divides both $ac+bd$ and $ad - bc$. In terms of $n$, determine the size of the largest possible set $S$.
number theory gaussian integer divisibility gaussian modular arithmetic proof by contradiction