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4

Part of 1999 Abels Math Contest (Norwegian MO)

Problems(1)

sum of alternating sums of a subset

Source: Norwegian Mathematical Olympiad 1999 - Abel Competition p4

2/11/2020
For every nonempty subset RR of S={1,2,...,10}S = \{1,2,...,10\}, we define the alternating sum A(R)A(R) as follows: If r1,r2,...,rkr_1,r_2,...,r_k are the elements of RR in the increasing order, then A(R)=rk−rk−1+rk−2−...+(−1)k−1r1A(R) = r_k -r_{k-1} +r_{k-2}- ... +(-1)^{k-1}r_1. (a) Is it possible to partition SS into two sets having the same alternating sum? (b) Determine the sum ∑RA(R)\sum_{R} A(R), where RR runs over all nonempty subsets of SS.
SumSubsetsalgebracombinatorics