Question 128
Let $m, n$ be positive integers with $m \geq n$. Let $S$ be the set of pairs $(a, b)$ of relatively prime positive integers such that $a, b \leq m$ and $a+b\gt m$.
For each pair $(a, b) \in S$, consider the nonnegative integer solution $(u, v)$ to the equation $a u-b v=n$ chosen with $v \geq 0$ 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 $(a, b) \in S$.
For each pair $(a, b) \in S$, consider the nonnegative integer solution $(u, v)$ to the equation $a u-b v=n$ chosen with $v \geq 0$ 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 $(a, b) \in S$.