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the number of sequences

Source: 2010 HongKong mathematic Olympiad

February 11, 2012
inequalitiescombinatorics proposedcombinatorics

Problem Statement

Let nn be a positive integer. Find the number of sequences x1,x2,…x2n−1,x2nx_{1},x_{2},\ldots x_{2n-1},x_{2n}, where xi∈{−1,1}x_{i}\in\{-1,1\} for each ii, satisfying the following condition: for any integer kk and mm such that 1≤k≤m≤n1\le k\le m\le n then the following inequality holds ∣∑i=2k−12mxi∣≤ 2\left|\sum_{i=2k-1}^{2m}x_{i}\right|\le\ 2
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