Difference between revisions of "User:Temperal/The Problem Solver's Resource2"

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===Trivial Inequality===
 
===Trivial Inequality===
For any real <math>x</math>, <math>x^2\ge 0</math>, with equality iff <math>x=0</math><math>.
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For any real <math>x</math>, <math>x^2\ge 0</math>, with equality iff <math>x=0</math>.
 
===Arithmetic Mean/Geometric Mean Inequality===
 
===Arithmetic Mean/Geometric Mean Inequality===
For any set of real numbers </math>S<math>, </math>\frac{S_1+S_2+S_3....+S_{k-1}+S_k}{k}\ge \sqrt[k]{S_1\cdot S_2 \cdot S_3....\cdot S_{k-1}\cdot S_k}<math> with equality iff </math>S_1=S_2=S_3...=S_{k-1}=S_k<math>.
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For any set of real numbers <math>S</math>, <math>\frac{S_1+S_2+S_3....+S_{k-1}+S_k}{k}\ge \sqrt[k]{S_1\cdot S_2 \cdot S_3....\cdot S_{k-1}\cdot S_k}</math> with equality iff <math>S_1=S_2=S_3...=S_{k-1}=S_k</math>.
  
  
 
===Cauchy-Schwarz inequality===
 
===Cauchy-Schwarz inequality===
  
For any real numbers </math>a_1,a_2,...,a_n<math> and </math>b_1,b_2,...,b_n<math>, the following holds:
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For any real numbers <math>a_1,a_2,...,a_n</math> and <math>b_1,b_2,...,b_n</math>, the following holds:
  
</math>\displaystyle(\sum a_i^2)(\sum b_i^2) \ge (\sum a_ib_i)^2<math>
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<math>(\sum a_i^2)(\sum b_i^2) \ge (\sum a_ib_i)^2</math>
  
 
====Cauchy-Schwarz variation====
 
====Cauchy-Schwarz variation====
  
For any real numbers </math>a_1,a_2,...,a_n<math> and positive real numbers </math>b_1,b_2,...,b_n<math>, the following holds:
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For any real numbers <math>a_1,a_2,...,a_n</math> and positive real numbers <math>b_1,b_2,...,b_n</math>, the following holds:
  
</math>\displaystyle\sum\left({{a_i^2}\over{b_i}}\right) \ge {{\sum a_i^2}\over{\sum b_i}}$.
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<math>\sum\left({{a_i^2}\over{b_i}}\right) \ge {{\sum a_i^2}\over{\sum b_i}}</math>.
  
  
 
[[User:Temperal/The Problem Solver's Resource1|Back to page 1]] | [[User:Temperal/The Problem Solver's Resource2|Continue to page 3]]
 
[[User:Temperal/The Problem Solver's Resource1|Back to page 1]] | [[User:Temperal/The Problem Solver's Resource2|Continue to page 3]]
 
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Revision as of 15:52, 29 September 2007



The Problem Solver's Resource
Introduction | Other Tips and Tricks | Methods of Proof | You are currently viewing page 2.

Simple Number Theory

This is a collection of essential AIME-level number theory theorems and other tidbits.

Trivial Inequality

For any real $x$, $x^2\ge 0$, with equality iff $x=0$.

Arithmetic Mean/Geometric Mean Inequality

For any set of real numbers $S$, $\frac{S_1+S_2+S_3....+S_{k-1}+S_k}{k}\ge \sqrt[k]{S_1\cdot S_2 \cdot S_3....\cdot S_{k-1}\cdot S_k}$ with equality iff $S_1=S_2=S_3...=S_{k-1}=S_k$.


Cauchy-Schwarz inequality

For any real numbers $a_1,a_2,...,a_n$ and $b_1,b_2,...,b_n$, the following holds:

$(\sum a_i^2)(\sum b_i^2) \ge (\sum a_ib_i)^2$

Cauchy-Schwarz variation

For any real numbers $a_1,a_2,...,a_n$ and positive real numbers $b_1,b_2,...,b_n$, the following holds:

$\sum\left({{a_i^2}\over{b_i}}\right) \ge {{\sum a_i^2}\over{\sum b_i}}$.


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