# Difference between revisions of "Arithmetic Mean-Geometric Mean Inequality"

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The AM-GM inequalitiy is a specific case of the [[Power mean inequality]]. It (and the much more general Power Mean Inequality) are used fairly frequently to solve Olympiad-level Inequality problems, such as those on the [[United States of America Mathematics Olympiad | USAMO]] and [[International Mathematics Olympiad | IMO]]. | The AM-GM inequalitiy is a specific case of the [[Power mean inequality]]. It (and the much more general Power Mean Inequality) are used fairly frequently to solve Olympiad-level Inequality problems, such as those on the [[United States of America Mathematics Olympiad | USAMO]] and [[International Mathematics Olympiad | IMO]]. | ||

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+ | === See also === | ||

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+ | * [[Algebra]] | ||

+ | * [[Inequalities]] | ||

+ | * [http://www.mathideas.org/problems/2006/5/29.pdf Basic Inequalities by Adeel Khan] | ||

+ | * [http://www.mathideas.org/problems/2006/5/31.pdf Inequalities: An Application of RMS-AM-GM-HM] |

## Revision as of 13:00, 21 June 2006

The **Arithmetic Mean-Geometric Mean** (**AM-GM**) Inequality states that the arithmetic mean of a set of positive real numbers is greater than or equal to the geometric mean of the same set of positive real numbers. For example, for the set , the Arithmetic Mean, 25, is greater than the Geometric Mean, 18; AM-GM guarantees this is always the case.

In general, AM-GM states that for a set of positive real numbers , the following always holds:

The AM-GM inequalitiy is a specific case of the Power mean inequality. It (and the much more general Power Mean Inequality) are used fairly frequently to solve Olympiad-level Inequality problems, such as those on the USAMO and IMO.