An algebra problem by Rohan K

Algebra Level 5

Let \(x_1, x_2, \ldots , x_{100} \) be non-negative real numbers such that \(x_i + x_{i+1} + x_{i+2} \leq 1\) for all \(i = 1,2,\ldots,100 \), where \(x_{101} = x_1 \) and \(x_{102} = x_2\).

Find the maximal possible value for the sum \( \displaystyle S = \sum_{i=1}^{100} x_i x_{i+2} \).

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