VMO 1993 Problem 5

1993 points are arranged in a circle. At time 0 each point is arbitrarily labeled +1 or -1. At times n = 1, 2, 3, ... the vertices are relabeled. At time n a vertex is given the label +1 if its two neighbours had the same label at time n-1, and it is given the label -1 if its two neighbours had different labels at time n-1. Show that for some time n > 1 the labeling will be the same as at time 1.