ISL 2007 A6

Answer

Let $a_{1}, a_{2}, \ldots, a_{100}$ be nonnegative real numbers such that $a_{1}^{2}+a_{2}^{2}+\ldots+a_{100}^{2}=1$ . Prove that $a_{1}^{2} a_{2}+a_{2}^{2} a_{3}+\ldots+a_{100}^{2} a_{1}<\frac{12}{25}$ (Poland)

Dataset text NuminaMath olympiads. Quoted with source.

Official text: IMO 2007 Shortlist booklet, page 21.

Show the answer

S \leq \sqrt{\frac{5+2 \sqrt{7}}{33+6 \sqrt{7}}} \approx 0.458879

Solutions

Official solution Solution 1 by the IMO Problem Selection Committee

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