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

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− | ===Cauchy-Schwarz | + | ===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: | 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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<math>(\sum a_i^2)(\sum b_i^2) \ge (\sum a_ib_i)^2</math> | <math>(\sum a_i^2)(\sum b_i^2) \ge (\sum a_ib_i)^2</math> | ||

− | ====Cauchy-Schwarz | + | ====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: | 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: |

## Revision as of 15:54, 29 September 2007

## Simple Number TheoryThis is a collection of essential AIME-level number theory theorems and other tidbits. ## Trivial InequalityFor any real , , with equality iff . ## Arithmetic Mean/Geometric Mean InequalityFor any set of real numbers , with equality iff .
## Cauchy-Schwarz InequalityFor any real numbers and , the following holds:
## Cauchy-Schwarz VariationFor any real numbers and positive real numbers , the following holds: . |