The maximum value of
Let be a set of positive integers not exceeding 24. Find the maximum value of so that have sums of all subsets different.
Example: If , then the possible sums of all subsets (including empty set) are , which are all different. In this case .
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1 solution
Suppose the maximum value of ∣ A ∣ = n ≥ 7 . From these n elements, we consider the sums of subsets at most 4 elements. There are at least ( 1 n ) + ( 2 n ) + ( 3 n ) + ( 4 n ) ≥ ( 1 7 ) + ( 2 7 ) + ( 3 7 ) + ( 4 7 ) = 9 8 such subsets. On the other hand, the maximum sum of the four distinct integers not exceeding 24 is 2 4 + 2 3 + 2 2 + 2 1 = 9 0 < 9 8 . Thus there are at least two of the subsets have the same sum. Hence the maximum value of ∣ A ∣ = n ≤ 6 .
The answer is 6 and one such possible set A is { 1 1 , 1 7 , 2 0 , 2 2 , 2 3 , 2 4 } .