2001 IMO Shortlist Problems/C6
For a positive integer define a sequence of zeros and ones to be balanced if it contains zeros and ones. Two balanced sequences and are neighbors if you can move one of the symbols of to another position to form . For instance, when , the balanced sequences and are neighbors because the third (or fourth) zero in the first sequence can be moved to the first or second position to form the second sequence. Prove that there is a set of at most balanced sequences such that every balanced sequence is equal to or is a neighbor of at least one sequence in .
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