OPM 2020 Problem 3

Given a subset of $\{1,2,...,n\}$ , we define its alternating sum in the following way: we order the elements of the subset in descending order and, starting with the largest, we alternately add and subtract the successive numbers. For example, the alternating sum of the set $\{1,3,4,6,8\}$ is $8-6+4-3+1 = 4$ . Determines the sum of the alternating sums of all subsets of $\{1,2,...,10\}$ with an odd number of elements.