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Maximize sum over x_i*x_{i+2}

Source: IMO Shortlist 2010, Algebra 3

July 17, 2011
inequalitiesIMO Shortlistn-variable inequality

Problem Statement

Let x1,…,x100x_1, \ldots , x_{100} be nonnegative real numbers such that xi+xi+1+xi+2≤1x_i + x_{i+1} + x_{i+2} \leq 1 for all i=1,…,100i = 1, \ldots , 100 (we put x101=x1,x102=x2).x_{101 } = x_1, x_{102} = x_2). Find the maximal possible value of the sum S=∑i=1100xixi+2.S = \sum^{100}_{i=1} x_i x_{i+2}.
Proposed by Sergei Berlov, Ilya Bogdanov, Russia
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