Let
![x_1, \ldots , x_{100}](/media/m/d/a/4/da4d451d69311167347d12579e12debc.png)
be nonnegative real numbers such that
![x_i + x_{i+1} + x_{i+2} \leq 1](/media/m/0/1/d/01d3df6f5178872838dcef5e43f59989.png)
for all
![i = 1, \ldots , 100](/media/m/4/0/6/406ad85c32dbf74d6e437e06724299a2.png)
(we put
![x_{101 } = x_1, x_{102} = x_2).](/media/m/5/2/9/5295243cbc6666f95bd50cfee0aa03fd.png)
Find the maximal possible value of the sum
![S = \sum^{100}_{i=1} x_i x_{i+2}.](/media/m/b/e/c/bec6bf414ef65cb8d2ea7e39472adf25.png)
Proposed by Sergei Berlov, Ilya Bogdanov, Russia
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Let $x_1, \ldots , x_{100}$ be nonnegative real numbers such that $x_i + x_{i+1} + x_{i+2} \leq 1$ for all $i = 1, \ldots , 100$ (we put $x_{101 } = x_1, x_{102} = x_2).$ Find the maximal possible value of the sum $S = \sum^{100}_{i=1} x_i x_{i+2}.$
Proposed by Sergei Berlov, Ilya Bogdanov, Russia