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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 $S$.

1997 Romania National Olympiad, 1

Tags: superior algebra , rings , units , gaussian integer

Let $\alpha \in \mathbb{C} \setminus \mathbb{Q}$ be such that the set $A= \{ a+b \alpha : a,b \in \mathbb{Z} \}$ is a ring with respect to the usual operations of $\mathbb{C}.$ If the ring $A$ has exactly four invertible elements, prove that $A= \mathbb{Z}[i].$

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