2001 IMO Shortlist Problems/C6
Problem
For a positive integer 
define a sequence of zeros and ones to be [i]balanced[/i] if it contains zeros and ones. Two balanced sequences 
and 
are [i]neighbors[/i] 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 .
Solution
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