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vertices of a 2n-gon are labelled with the numbers 1,2,...,2n

Source: 1999 German Federal - Bundeswettbewerb Mathematik - BWM - Round 2 p1

January 27, 2020
combinatorial geometrycombinatorics

Problem Statement

The vertices of a regular 2n2n-gon (with n>2n > 2 an integer) are labelled with the numbers 1,2,...,2n1,2,...,2n in some order. Assume that the sum of the labels at any two adjacent vertices equals the sum of the labels at the two diametrically opposite vertices. Prove that this is possible if and only if nn is odd.
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