Equal sum?

Given such a set $S$ , there exist $2^{10} - 2 = 1022$ different non-trivial subsets. The maximum sum in such a subset is $91 + \cdots + 99 = 855$ ; the minimum sum is $10$ . Thus we have a mapping $\Sigma:\ \mathcal P(S) - \{\emptyset,S\} \to \{10, \cdots, 945\}.$ Since the codomain is strictly smaller than the domain, this function cannot be injective. There exist distinct $A, B \subset S$ such that $\Sigma(A) = \Sigma(B)$ .

These sets are not necessarily disjoint. However, we may remove common elements: $A' = A - (A \cap B)$ and $B' = B - (A \cap B)$ . It is obvious that $\Sigma(A') = \Sigma(B')$ , that they are not empty, and that they are disjoint.

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Note that there exists a set of seven 2-digit numbers for which this property does not hold; for instance, $S = \{ 16,17,24,32,34,64,68 \}$ .

I don't think there are any sets of eight or nine 2-digit numbers for which this is true. However, $2^9 - 2 = 510 < 764 = 92 + \cdots + 100$ , so that the reasoning above won't work.