Difference between revisions of "Inequality"
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We say that ''a>b'' (or, equivalently, ''b<a'') if ''a'' and ''b'' are [[real number]]s, and ''a-b'' is a [[positive number]]. However, there are many inequalities that are much more interesting and also very important, such as the ones listed below. | We say that ''a>b'' (or, equivalently, ''b<a'') if ''a'' and ''b'' are [[real number]]s, and ''a-b'' is a [[positive number]]. However, there are many inequalities that are much more interesting and also very important, such as the ones listed below. | ||
| − | |||
| − | + | ==Introductory== | |
* [[AM-GM]] for 2 variables | * [[AM-GM]] for 2 variables | ||
* [[Geometric inequalities]] | * [[Geometric inequalities]] | ||
* [[Trivial Inequality]] | * [[Trivial Inequality]] | ||
| − | + | ||
| − | === | + | ==Intermediate== |
| − | * [[ | + | === Example Problems === |
| + | * [[1992_AIME_Problems/Problem_3 | 1992 AIME Problem 3]] | ||
| + | |||
| + | |||
| + | ==Olympiad== | ||
| + | See the list of famous inequalities below | ||
Revision as of 20:24, 26 July 2006
The subject of mathematical inequalities is tied closely with optimization methods. While most of the subject of inequalities is often left out of the ordinary educational track, they are common in mathematics Olympiads.
Contents
Motivation
We say that a>b (or, equivalently, b<a) if a and b are real numbers, and a-b is a positive number. However, there are many inequalities that are much more interesting and also very important, such as the ones listed below.
Introductory
- AM-GM for 2 variables
- Geometric inequalities
- Trivial Inequality
Intermediate
Example Problems
Olympiad
See the list of famous inequalities below
Famous inequalities
Here are some of the more famous and useful inequalities, as well as general inequalities topics.
- Arithmetic Mean-Geometric Mean Inequality
- Cauchy-Schwarz Inequality
- Chebyshev's Inequality
- Geometric inequalities
- Hölder's inequality
- Isoperimetric inequalities
- Jensen's Inequality
- Minkowski Inequality
- Muirhead's Inequality
- Power mean inequality
- Rearrangement Inequality
- Schur's Inequality
- Triangle Inequality
- Trigonometric inequalities
- Trivial inequality
Problem solving tactics
substitution, telescoping, induction, etc. (write me please!)
Resources
Books
Intermediate
Olympiad
- The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities by J. Michael Steele.
- Problem Solving Strategies by Arthur Engel contains significant material on inequalities.
- Inequalities by G. H. Hardy, J. E. Littlewood, G. Pólya.
Articles
Olympiad
- Inequalities by MIT Professor Kiran Kedlaya.
- Inequalities by IMO gold medalist Thomas Mildorf.
Classes
Olympiad
- The Worldwide Online Olympiad Training Program is designed to help students learn to tackle mathematical Olympiad problems in topics such as inequalities.
