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red points among 20 collinear points

Source: XII May Olympiad (Olimpiada de Mayo) 2006 L1 P3

September 22, 2022
Coloringcombinatorics

Problem Statement

There are 2020 collinear points, separated by the same distance: .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .. \,\,\, . \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\, . Miguel has to paint three or more of these points red, in such a way that the red points are separated by the same distance and it is impossible to paint exactly one more point red without violating the previous condition. Determine in how many ways Miguel can do his homework.
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