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"odd" sets and "even" sets

Source: Romania TST 1994

August 28, 2009
combinatorics proposedcombinatorics

Problem Statement

Let X_n\equal{}\{1,2,...,n\},where n≥3 n \geq 3. We define the measure m(X) m(X) of X⊂Xn X\subset X_n as the sum of its elements.(If |X|\equal{}0,then m(X)\equal{}0). A set X⊂Xn X \subset X_n is said to be even(resp. odd) if m(X) m(X) is even(resp. odd). (a)Show that the number of even sets equals the number of odd sets. (b)Show that the sum of the measures of the even sets equals the sum of the measures of the odd sets. (c)Compute the sum of the measures of the odd sets.
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