Arithmetic Mean-Geometric Mean Inequality
The Arithmetic Mean-Geometric Mean (often abbreviated 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.