1993 USAMO Problems/Problem 5

Revision as of 15:29, 15 April 2012 by Danielguo94 (talk | contribs)

Problem 5

Let $a_0, a_1, a_2,\cdots$ be a sequence of positive real numbers satisfying $a_{i-1}a_{i+1}\le a^2_i$ for $i = 1, 2, 3,\cdots$ . (Such a sequence is said to be log concave.) Show that for each $n > 1$,

$\frac{a_0+\cdots+a_n}{n+1}\cdot\frac{a_1+\cdots+a_{n-1}}{n-1}\ge\frac{a_0+\cdots+a_{n-1}}{n}\cdot\frac{a_1+\cdots+a_{n}}{n}$.

Solution

Resources

1993 USAMO (ProblemsResources)
Preceded by
Problem 4
Followed by
Last Problem
1 2 3 4 5
All USAMO Problems and Solutions
Invalid username
Login to AoPS