Problem

Source: China Northern MO 2009 p7 CNMO

Tags: number theory, algebra, floor function



Let $\lfloor m \rfloor$ be the largest integer smaller than $m$ . Assume $x,y \in \mathbb{R+}$ , For all positive integer $n$ , $\lfloor x \lfloor ny \rfloor \rfloor =n-1$ . Prove : $xy=1$ , $y$ is an irrational number larger than $ 1 $ .