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Found problems: 426

1993 USAMO, 5
Tags: inequalities , induction , inequalities unsolved
Let $ \, a_{0}, a_{1}, a_{2},\ldots\,$ be a sequence of positive real numbers satisfying $ \, a_{i\minus{}1}a_{i\plus{}1}\leq a_{i}^{2}\,$ for $ i \equal{} 1,2,3,\ldots\; .$ (Such a sequence is said to be [i]log concave[/i].) Show that for each $ \, n > 1,$ \[ \frac{a_{0}\plus{}\cdots\plus{}a_{n}}{n\plus{}1}\cdot\frac{a_{1}\plus{}\cdots\plus{}a_{n\minus{}1}}{n\minus{}1}\geq\frac{a_{0}\plus{}\cdots\plus{}a_{n\minus{}1}}{n}\cdot\frac{a_{1}\plus{}\cdots\plus{}a_{n}}{n}.\]