MathDB
Operations on n-tuples

Source: 2009 IrMO Paper 1 Problem 4

January 30, 2018
combinatorics

Problem Statement

Given an nn-tuple of numbers (x1,x2,…,xn)(x_1,x_2,\dots ,x_n) where each xi=+1x_i=+1 or −1-1, form a new nn-tuple (x1x2,x2x3,x3x4,…,xnx1),(x_1x_2,x_2x_3,x_3x_4,\dots ,x_nx_1), and continue to repeat this operation. Show that if n=2kn=2^k for some integer k≥1k\ge 1, then after a certain number of repetitions of the operation, we obtain the nn-tuple (1,1,1,…,1).(1,1,1,\dots ,1).
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