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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 inequality courses.
| + | #REDIRECT[[AM-GM Inequality]] |
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| − | == Theorem ==
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| − | The AM-GM states that for any [[set]] of [[positive]] [[real number]]s, the [[arithmetic mean]] of the set is greater than or [[equal]] to the [[geometric mean]] of the set. Algebraically, this is written:
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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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| − | <center><math>\frac{a_1+a_2+\ldots+a_n}{n}\geq\sqrt[n]{a_1a_2\cdots a_n}</math></center>
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| − | Using the shorthand notation for [[summation]]s and [[product]]s:
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| − | <center><math>\frac{\sum\limits_{i=1}^{n}a_i}{n}\geq\sqrt[n]{\prod\limits_{i=1}^{n}a_i}</math></center>
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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 arithmetic mean and geometric mean are equal [[if and only if]] all members of the set are equal.
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| − | AMGM 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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| − | === Proof ===
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| − | {{incomplete|proof}}
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| − | === Weighted Form ===
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| − | The weighted form of AMGM is given by using [[weighted average]]s. For example, the weighted arithmetic mean of <math>x</math> and <math>y</math> with <math>3:1</math> is <math>\frac{3y+x}{2\cdot 3}</math> and the geometric is <math>\sqrt[2+3]{xy^3}</math>. AMGM applies to weighted averages.
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| − | ==Extensions==
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| − | *The [[power mean inequality]] is a useful inequality extending on the arithmetic mean side of the inequality.
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| − | *The [[root-square-mean arithmetic-mean geometric-mean harmonic-mean inequality]] is an extension on both sides of the inequality with further types of means.
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| − | ==Problems==
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| − | === Introductory ===
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| − | === Intermediate ===
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| − | * Find the minimum value of <math>\frac{9x^2\sin^2 x + 4}{x\sin x}</math> for <math>0 < x < \pi</math>.
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| − | ([[1983 AIME Problems/Problem 9|Source]])
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| − | === Olympiad ===
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| − | * Let <math>a </math>, <math>b </math>, and <math>c </math> be positive real numbers. Prove that
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| − | <center>
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| − | <math>
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| − | (a^5 - a^2 + 3)(b^5 - b^2 + 3)(c^5 - c^2 + 3) \ge (a+b+c)^3
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| − | </math>.
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| − | </center>
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| − | ([[2004 USAMO Problems/Problem 5|Source]])
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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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| − | ==External Links==
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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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| − | [[Category:Inequality]]
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| − | [[Category:Theorems]]
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