Problem

Source: Romania National Olympiad 2023

Tags: floor function, number theory



Let $r$ and $s$ be real numbers in the interval $[1, \infty)$ such that for all positive integers $a$ and $b$ with $a \mid b \implies \left\lfloor ar \right\rfloor$ divides $\left\lfloor bs \right\rfloor$. a) Prove that $\frac{s}{r}$ is a natural number. b) Show that both $r$ and $s$ are natural numbers. Here, $\lfloor x \rfloor$ denotes the greatest integer that is less than or equal to $x$.