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$1993$ points are arranged in a circle

Source: 31-th Vietnamese Mathematical Olympiad 1993

February 17, 2007
combinatorics unsolvedcombinatorics

Problem Statement

19931993 points are arranged in a circle. At time 00 each point is arbitrarily labeled +1+1 or −1-1. At times n=1,2,3,...n = 1, 2, 3, ... the vertices are relabeled. At time nn a vertex is given the label +1+1 if its two neighbours had the same label at time n−1n-1, and it is given the label −1-1 if its two neighbours had different labels at time n−1n-1. Show that for some time n>1n > 1 the labeling will be the same as at time 1.1.
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