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Sum of differences over 2n-1 consecutive terms is one

Source: CWMO 2003, Problem 2

December 27, 2008
vectorinequalities

Problem Statement

Let a1,a2,…,a2n a_1, a_2, \ldots, a_{2n} be 2n 2n real numbers satisfying the condition \sum_{i \equal{} 1}^{2n \minus{} 1} (a_{i \plus{} 1} \minus{} a_i)^2 \equal{} 1. Find the greatest possible value of (a_{n \plus{} 1} \plus{} a_{n \plus{} 2} \plus{} \ldots \plus{} a_{2n}) \minus{} (a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n).
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