NO-ABEL 1999 Problem 4

For every nonempty subset $R$ of $S = \{1,2,...,10\}$ , we define the alternating sum $A(R)$ as follows: If $r_1,r_2,...,r_k$ are the elements of $R$ in the increasing order, then $A(R) = r_k -r_{k-1} +r_{k-2}- ... +(-1)^{k-1}r_1$ . (a) Is it possible to partition $S$ into two sets having the same alternating sum? (b) Determine the sum $\sum_{R} A(R)$ , where $R$ runs over all nonempty subsets of $S$ .