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Partitioning the set of binary sequence with a condition

Source: Bulgarian National Olympiad 2012 Problem 4

May 21, 2012
inductionabstract algebragroup theorycombinatorics proposedcombinatorics

Problem Statement

Let nn be an even natural number and let AA be the set of all non-zero sequences of length nn, consisting of numbers 00 and 11 (length nn binary sequences, except the zero sequence (0,0,…,0)(0,0,\ldots,0)). Prove that AA can be partitioned into groups of three elements, so that for every triad {(a1,a2,…,an),(b1,b2,…,bn),(c1,c2,…,cn)}\{(a_1,a_2,\ldots,a_n), (b_1,b_2,\ldots,b_n), (c_1,c_2,\ldots,c_n)\}, and for every i=1,2,…,ni = 1, 2,\ldots,n, exactly zero or two of the numbers ai,bi,cia_i, b_i, c_i are equal to 11.
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