Problem of the Month: June
At the beginning of each month, a new problem is posted here; solutions are posted
around the end of the month.
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Problem of the Month - June 2010: S-S-S-Sum!
For a given set A, let the S-value of A denote the sum of the
odd-numbered elements (1st, 3rd, 5th, etc.) minus the sum of the even-numbered elements
(2nd, 4th, etc.) after A has been sorted in non-increasing order;
in otherwords, alternate numbers are added or subtracted from the total sum. For
example, the S-value of {1,4,6,4,8,3,7} is the S-value of {8,7,6,4,4,3,1}
= 8 - 7 + 6 - 4 + 4 - 3 + 1 = 5.
For all proper subsets of {1,2,3,4,5,6,7}, there exists a positive S-value
associated with the subset. Determine the sum of all of these S-values.