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Maximize a sum

Source: Austrian-Polish 2001

September 19, 2006
inequalities unsolvedinequalities

Problem Statement

The sequence a1,a2,⋯ ,a2010a_{1},a_{2},\cdots,a_{2010} has the following properties: (1) each sum of the 20 successive values of the sequence is nonnegative, (2) ∣aiai+1∣≤1|a_{i}a_{i+1}| \leq 1 for i=1,2,⋯ ,2009i=1,2,\cdots,2009. Determine the maximal value of the expression ∑i=12010ai\sum_{i=1}^{2010}a_{i}.
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