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IMC 2021, first day , problem 3

Source: IMC , first day problem 3

August 4, 2021
number theoryIMC 2021

Problem Statement

We say that a positive real number dd is goodgood if there exists an infinite squence a1,a2,a3,...∈(0,d)a_1,a_2,a_3,...\in (0,d) such that for each nn, the points a1,a2,...,ana_1,a_2,...,a_n partition the interval [0,d][0,d] into segments of length at most 1n\frac{1}{n} each . Find sup{d∣dis good}\text{sup}\{d| d \text{is good}\}.
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