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min k such among k reals, exist a,b in R of which |a -b|<1/n or |a-b| >n

Source: The Francophone MO Juniors P3

August 10, 2020
combinatoricsalgebraFrancophone

Problem Statement

Let nn be an integer greater than or equal to 11. Find, as a function of nn, the smallest integer k≥2k\ge 2 such that, among any kk real numbers, there are necessarily two of which the difference, in absolute value, is either strictly less than 1/n1 / n, either strictly greater than nn.
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