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# Problem 105

## Special subset sums: testing

Let $S(A)$ represent the sum of elements in set $A$ of size $n$. We shall call it a special sum set if for any two non-empty disjoint subsets, $B$ and $C$, the following properties are true:

1. $S(B)\neq S(C)$; that is, sums of subsets cannot be equal.
2. If $B$ contains more elements than $C$ then $S(B) > S(C)$.

For example, $\{81, 88, 75, 42, 87, 84, 86, 65\}$ is not a special sum set because $65 + 87 + 88 = 75 + 81 + 84$, whereas $\{157, 150, 164, 119, 79, 159, 161, 139, 158\}$ satisfies both rules for all possible subset pair combinations and $S(A) = 1286$.

Using sets.txt (right click and “Save Link/Target As…”), a 4K text file with one-hundred sets containing seven to twelve elements (the two examples given above are the first two sets in the file), identify all the special sum sets, $A_1, A_2, \ldots , A_k$, and find the value of $S(A_1) + S(A_2) +\ldots + S(A_k)$.

NOTE: This problem is related to Problem 103 and Problem 106.

## 特殊的子集和：检验

1. $S(B) \neq S(C)$；也就是说，任意子集的和不相同。
2. 如果$B$中的元素比$C$多，则$S(B) > S(C)$。