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]]
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* [[Inequalities]]
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* [http://www.mathideas.org/problems/2006/5/29.pdf Basic Inequalities by Adeel Khan]
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* [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 $\{9,12,54\}$, 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 $a_1,a_2,\ldots,a_n$, the following always holds:

$\displaystyle\left(\frac{a_1+a_2+\ldots+a_n}{n}\right)\geq\sqrt[n]{a_1a_2\cdots a_n}$

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.

See also

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