Suppose S is a set of nine distinct integers whose sum is greater than 200
with no 4-element subset whose sum is greater than 100. Let A be the set
consisting of the four largest elements of S. The smallest element of A
can be no bigger than 23, because 24+25+26+27>100. So the five smallest
elements of S range in size from 1 to 22. The sum of the five smallest
elements of S could not be bigger than 18+19+20+21+22=100, and the sum of the
four largest elements are not bigger than 100, so the sum of the nine elements
of S is not greater than 200, a contradiction.
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