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| − | The '''Arithmetic Mean-Geometric Mean Inequality''' ('''AM-GM''' or '''AMGM''') is an elementary inequality, generally one of the first ones taught in number theory courses.
| + | #REDIRECT[[AM-GM Inequality]] |
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| − | == Inequality ==
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| − | The AM-GM states that for any multiset of positive real numbers, the arithmetic mean of the set is greater than or equal to the geometric mean of the set. Or:
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| − | For a set of nonnegative real numbers <math>a_1,a_2,\ldots,a_n</math>, the following always holds:
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| − | <math>\left(\frac{a_1+a_2+\ldots+a_n}{n}\right)\geq\sqrt[n]{a_1a_2\cdots a_n}</math>
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| − | For example, for the set <math>\{9,12,54\}</math>, the Arithmetic Mean, 25, is greater than the Geometric Mean, 18; AM-GM guarantees this is always the case.
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| − | The [[equality condition]] of this [[inequality]] states that the AM and GM are equal if and only if all members of the set are equal.
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| − | The AM-GM inequalitiy is a specific case of the [[power mean inequality]]. Both can be 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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| − | * [[RMS-AM-GM-HM]]
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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 by Adeel Khan]
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| − | {{wikify}}
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| − | {{stub}}
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| − | [[Category:Number theory]]
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| − | [[Category:Theorems]]
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| − | [[Category:Algebra]]
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