Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 5.1 AIME, harder Prove it Belarus

Let $x_1, \ldots, x_{100}$ be nonnegative real numbers such that $x_i + x_{i+1} + x_{i+2} \le 1$ for all $i = 1, \ldots, 100$ (we put $x_{101} = x_1, x_{102} = x_2$ ).
Find the maximal possible value of the sum $S = \sum_{i=1}^{100} x_i x_{i+2}$ .

Solution

3. See IMO-2010 Shortlist, Problem A3.

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