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ASU 283 All Soviet Union MO 1979 n points on a line

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July 14, 2019
combinatoricspointscollinear

Problem Statement

Given nn points (in sequence)A1,A2,...,An A_1, A_2, ... , A_n on a line. All the segments A1A2A_1A_2, A2A3A_2A_3,... ..., An−1AnA_{n-1}A_n are shorter than 11. We need to mark (k−1)(k-1) points so that the difference of every two segments, with the ends in the marked points, is shorter than 11. Prove that it is possible
a) for k=3k=3,
b) for every kk less than (n−1)(n-1).
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