Found problems: 1
Let $m,n$ be positive integers with $m \ge n$. Let $S$ be the set of pairs $(a,b)$ of relatively prime positive integers such that $a,b \le m$ and $a+b > m$.
For each pair $(a,b)\in S$, consider the nonnegative integer solution $(u,v)$ to the equation $au - bv = n$ chosen with $v \ge 0$ minimal, and let $I(a,b)$ denote the (open) interval $(v/a, u/b)$.
Prove that $I(a,b) \subseteq (0,1)$ for every $(a,b)\in S$, and that any fixed irrational number $\alpha\in(0,1)$ lies in $I(a,b)$ for exactly $n$ distinct pairs $(a,b)\in S$.
[i]Victor Wang, inspired by 2013 ISL N7[/i]