Problem

Source: Cyprus 2022 Junior TST-2 Problem 4

Tags: number theory, pigeonhole principle



Let $A$ be a subset of $\{1, 2, 3, \ldots, 50\}$ with the property: for every $x,y\in A$ with $x\neq y$, it holds that \[\left| \frac{1}{x}- \frac{1}{y}\right|>\frac{1}{1000}.\]Determine the largest possible number of elements that the set $A$ can have.