

(29 intermediate revisions by 15 users not shown) 
Line 1: 
Line 1: 
−  The '''Arithmetic MeanGeometric Mean Inequality''' ('''AMGM''' or '''AMGM''') is an elementary inequality, generally one of the first ones taught in number theory courses.
 +  #REDIRECT[[AMGM Inequality]] 
−   
−  == Inequality ==
 
−  The AMGM 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:
 
−   
−  For a set of nonnegative real numbers <math>a_1,a_2,\ldots,a_n</math>, the following always holds:
 
−   
−  <math>\left(\frac{a_1+a_2+\ldots+a_n}{n}\right)\geq\sqrt[n]{a_1a_2\cdots a_n}</math>
 
−   
−  For example, for the set <math>\{9,12,54\}</math>, the Arithmetic Mean, 25, is greater than the Geometric Mean, 18; AMGM guarantees this is always the case.
 
−   
−  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.
 
−   
−  The AMGM inequalitiy is a specific case of the [[power mean inequality]]. Both can be used fairly frequently to solve Olympiadlevel Inequality problems, such as those on the [[United States of America Mathematics Olympiad  USAMO]] and [[International Mathematics Olympiad  IMO]].
 
−   
−   
−  == See also ==
 
−   
−  * [[RMSAMGMHM]]
 
−  * [[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 RMSAMGMHM by Adeel Khan]
 
−   
−  {{wikify}}
 
−   
−  {{stub}}
 
−  [[Category:Number theory]]
 
−  [[Category:Theorems]]
 
−  [[Category:Algebra]]
 