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$n$ points on a line, distance <1

Source: 2022 Grosman Mathematical Olympiad P7

September 22, 2022
combinatoricscombinatorics unsolved

Problem Statement

Let knk\leq n be two positive integers. nn points are marked on a line. It is known that for each marked point, the number of marked points at a distance 1\leq 1 from it (including the point itself) is divisible by kk. Show that kk divides nn (without remainder).