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Maximal subset with |1/x-1/y| >= 1/1000

Source: Cyprus 2022 Junior TST-2 Problem 4

February 21, 2022
number theorypigeonhole principle

Problem Statement

Let AA be a subset of {1,2,3,…,50}\{1, 2, 3, \ldots, 50\} with the property: for every x,y∈Ax,y\in A with x≠yx\neq y, it holds that ∣1x−1y∣>11000.\left| \frac{1}{x}- \frac{1}{y}\right|>\frac{1}{1000}. Determine the largest possible number of elements that the set AA can have.
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