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Pairs of points with same distance in convex n-gon

Source: 2014 China TST 2 Day 1 Q3

March 20, 2014
combinatorics proposedcombinatorics

Problem Statement

AA is the set of points of a convex nn-gon on a plane. The distinct pairwise distances between any 22 points in AA arranged in descending order is d1>d2>...>dm>0d_1>d_2>...>d_m>0. Let the number of unordered pairs of points in AA such that their distance is did_i be exactly μi\mu _i, for i=1,2,...,mi=1, 2,..., m. Prove: For any positive integer k≤mk\leq m, μ1+μ2+...+μk≤(3k−1)n\mu _1+\mu _2+...+\mu _k\leq (3k-1)n.
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