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ICMC 2019/20 Round 1, Problem 6

Source: Imperial College Mathematics Competition 2019/20 - Round 1

August 7, 2020
college contests

Problem Statement

Let ε<12\varepsilon < \frac{1}{2} be a positive real number and let UεU_{\varepsilon} denote the set of real numbers that differ from their nearest integer by at most ε\varepsilon. Prove that there exists a positive integer mm such that for any real number xx, the sets {x,2x,3x,...,mx}\left\{x, 2x, 3x, . . . , mx\right\} and UεU_{\varepsilon} have at least one element in common.
proposed by the ICMC Problem Committee
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