1997 OIM Problems/Problem 1

Problem

Let $r \ge 1$ be a real number that satisfies the following property:

For each pair of positive integers $m$ and $n$ , with $n$ multiple of $m$ we have that $\left\lfloor nr \right\rfloor$ is a multiple of $\left\lfloor mr \right\rfloor$ .

Prove that $r$ is an integer.

Note: If $x$ is a real number, we denote by $\left\lfloor x \right\rfloor$ the largest integer less than or equal to $x$ .

~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

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1997 OIM Problems/Problem 1

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