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WSUM = 3(a_1 + a_3 +..) + 2(a_2 + a_4 +...) , sum of WSUMs

Source: CRMO 2012 region 5 p6 Mumbai

September 30, 2018
Subsetsset theorycombinatoricsCombinatorics of set

Problem Statement

Let SS be the set {1,2,...,10}\{1, 2, ..., 10\}. Let AA be a subset of SS. We arrange the elements of AA in increasing order, that is, A={a1,a2,....,ak}A = \{a_1, a_2, ...., a_k\} with a1<a2<...<aka_1 < a_2 < ... < a_k. Define WSUM for this subset as 3(a1+a3+..)+2(a2+a4+...)3(a_1 + a_3 +..) + 2(a_2 + a_4 +...) where the first term contains the odd numbered terms and the second the even numbered terms. (For example, if A={2,5,7,8}A = \{2, 5, 7, 8\}, WSUM is 3(2+7)+2(5+8)3(2 + 7) + 2(5 + 8).) Find the sum of WSUMs over all the subsets of S. (Assume that WSUM for the null set is 00.)
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