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Difference between revisions of "Inequality"

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*If <math>a \ge b</math>, then <math>a+c>b</math>, where <math>c>0</math>.
 
*If <math>a \ge b</math>, then <math>a+c>b</math>, where <math>c>0</math>.
  
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==Common theorems==
 
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* The [[Arithmetic Mean-Geometric Mean Inequality]], its extension, the [[Power Mean Inequality]], and its other extension, the [[Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality]].
 
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* The [[Trivial Inequality]].
 
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* For others, consult [[User:Temperal/The Problem Solver's Resource4|this page]], [[User:Temperal/The Problem Solver's Resource8|this page]], and [[User:Temperal/The Problem Solver's Resource11|this page]].
 
  
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==Intermediate==
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== Theorems ==
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=== Example Problems ===
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Here are some of the more useful inequality theorems, as well as general inequality topics.
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* [[1992_AIME_Problems/Problem_3 | 1992 AIME Problem 3]]
 
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==Olympiad==
 
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See the list of famous inequalities below
 
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== Famous inequalities ==
 
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Here are some of the more famous and useful inequalities, as well as general inequalities topics.
 
 
* [[Arithmetic Mean-Geometric Mean | Arithmetic Mean-Geometric Mean Inequality]]
 
* [[Arithmetic Mean-Geometric Mean | Arithmetic Mean-Geometric Mean Inequality]]
 
* [[Cauchy-Schwarz Inequality]]
 
* [[Cauchy-Schwarz Inequality]]
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* [[Ptolemy's Inequality]]
 
* [[Ptolemy's Inequality]]
 
* [[Rearrangement Inequality]]
 
* [[Rearrangement Inequality]]
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* [[Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality]]
 
* [[Schur's Inequality]]
 
* [[Schur's Inequality]]
 
* [[Triangle Inequality]]
 
* [[Triangle Inequality]]
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* [[Trigonometric inequalities]]
 
 
* [[Trivial inequality]]
 
* [[Trivial inequality]]
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== Problem solving tactics ==
 
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substitution, telescoping, induction, etc. (write me please!)
 
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==Problems==
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===Introductory===
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*Given that <math>(a+1)(b+1)(c+1) = 8</math>, show that <math>abc \le 1</math>. (<url>weblog_entry.php?t=172070 Source</url>)
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===Intermediate===
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*A tennis player computes her win [[ratio]] by dividing the number of matches she has won by the total number of matches she has played. At the start of a weekend, her win ratio is exactly <math>.500</math>. During the weekend, she plays four matches, winning three and losing one. At the end of the weekend, her win ratio is greater than <math>.503</math>. What's the largest number of matches she could've won before the weekend began? ([[1992 AIME Problems/Problem 3|Source]])
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===Olympiad===
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*Let <math>a,b,c</math> be positive real numbers. Prove that
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<math>\frac{a}{\sqrt{a^{2}+8bc}}+\frac{b}{\sqrt{b^{2}+8ca}}+\frac{c}{\sqrt{c^{2}+8ab}}\ge 1</math> ([[2001 IMO Problems/Problem 2|Source]])
 
== Resources ==
 
== Resources ==
 
=== Books ===
 
=== Books ===

Revision as of 17:42, 25 October 2007

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.


Overview

Inequalities are arguably a branch of number theory. They deal with relations of variables denoted by four signs: $>,<,\ge,\le$.

For two numbers $a$ and $b$:

  • $a>b$ if $a$ is greater than $b$, that is, $a-b$ is positive.
  • $a<b$ if $a$ is smaller than $b$, that is, $a-b$ is negative.
  • $a\ge b$ if $a$ is greater than or equal to $b$, that is, $a-b$ is either positive or $0$.
  • $a\le b$ if $a$ is less than or equal to $b$, that is, $a-b$ is either negative or $0$.

Note that if and only if $a>b$, $b<a$, and vice versa. The same applies to the latter two signs: if and only if $a\ge b$, $b\le a$, and vice versa.

Some properties of inequalities are:

  • If $a>b$, then $a+c>b$, where $c\ge 0$.
  • If $a \ge b$, then $a+c\ge b$, where $c\ge 0$.
  • If $a \ge b$, then $a+c>b$, where $c>0$.


Theorems

Here are some of the more useful inequality theorems, as well as general inequality topics.


Problems

Introductory

  • Given that $(a+1)(b+1)(c+1) = 8$, show that $abc \le 1$. (<url>weblog_entry.php?t=172070 Source</url>)

Intermediate

  • A tennis player computes her win ratio by dividing the number of matches she has won by the total number of matches she has played. At the start of a weekend, her win ratio is exactly $.500$. During the weekend, she plays four matches, winning three and losing one. At the end of the weekend, her win ratio is greater than $.503$. What's the largest number of matches she could've won before the weekend began? (Source)

Olympiad

  • Let $a,b,c$ be positive real numbers. Prove that

$\frac{a}{\sqrt{a^{2}+8bc}}+\frac{b}{\sqrt{b^{2}+8ca}}+\frac{c}{\sqrt{c^{2}+8ab}}\ge 1$ (Source)

Resources

Books

Intermediate

Olympiad

Articles

Olympiad


Classes

Olympiad


See also

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