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black and blue vertices of a regular 20 -gon

Source: 2016 Saudi Arabia Pre-TST Level 4 2.2

September 13, 2020
combinatoricscombinatorial geometryColoring

Problem Statement

Ten vertices of a regular 2020-gon A1A2....A20A_1A_2....A_{20} are painted black and the other ten vertices are painted blue. Consider the set consisting of diagonal A1A4A_1A_4 and all other diagonals of the same length. 1. Prove that in this set, the number of diagonals with two black endpoints is equal to the number of diagonals with two blue endpoints. 2. Find all possible numbers of the diagonals with two black endpoints.
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