Problem

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Tags: number theory, algebra, floor function



Real numbers $x$ and $y$ not less than $1$ have the property that $$\left\lfloor\frac xy\right\rfloor=\frac{\lfloor nx\rfloor}{\lfloor ny\rfloor}\enspace\text{for any }n\in\mathbb N.$$Prove that either $x=y$ or $x$ and $y$ are integers, one dividing the other.