# Difference between revisions of "User:Temperal/The Problem Solver's Resource11"

(→Holder's Inequality) |
(→Mauclarin's Inequality: update) |
||

Line 29: | Line 29: | ||

with equality exactly iff all <math>x_i</math> are equivalent. | with equality exactly iff all <math>x_i</math> are equivalent. | ||

===Mauclarin's Inequality=== | ===Mauclarin's Inequality=== | ||

− | For non-negative real numbers <math>x_1,x_2,x_3 \ldots, x_n</math>, the following holds: | + | For non-negative real numbers <math>x_1,x_2,x_3 \ldots, x_n</math>, and <math>d_1,d_2,d_3 \ldots, d_n</math> such that |

+ | <cmath>d_k = \frac{\displaystyle \sum_{ 1\leq i_1 < i_2 < \cdots < i_k \leq n}x_{i_1} x_{i_2} \cdots x_{i_k}}{\displaystyle {n \choose k}}</cmath>, for <math>k\subset [1,n]</math> the following holds: | ||

− | <cmath> | + | <cmath>d_1 \ge \sqrt[2]{d_2} \ge \sqrt[3]{d_3}\ldots \ge \sqrt[n]{d_n}</cmath> |

with equality iff all <math>x_i</math> are equivalent. | with equality iff all <math>x_i</math> are equivalent. | ||

+ | |||

[[User:Temperal/The Problem Solver's Resource10|Back to page 10]] | Last page (But also see the | [[User:Temperal/The Problem Solver's Resource10|Back to page 10]] | Last page (But also see the | ||

[[User:Temperal/The Problem Solver's Resource Tips and Tricks|tips and tricks page]], and the | [[User:Temperal/The Problem Solver's Resource Tips and Tricks|tips and tricks page]], and the | ||

[[User:Temperal/The Problem Solver's Resource Competition|competition]]! | [[User:Temperal/The Problem Solver's Resource Competition|competition]]! | ||

|}<br /><br /> | |}<br /><br /> |

## Revision as of 11:30, 13 October 2007

## Advanced Number TheoryThese are Olympiad-level theorems and properties of numbers that are routinely used on the IMO and other such competitions. ## Jensen's InequalityFor a convex function and real numbers and , the following holds:
## Holder's InequalityFor positive real numbers , the following holds:
## Muirhead's InequalityFor a sequence that majorizes a sequence , then given a set of positive integers , the following holds:
## Rearrangement InequalityFor any multi sets and , is maximized when is greater than or equal to exactly of the other members of , then is also greater than or equal to exactly of the other members of . ## Newton's InequalityFor non-negative real numbers and the following holds: , with equality exactly iff all are equivalent. ## Mauclarin's InequalityFor non-negative real numbers , and such that , for the following holds:
with equality iff all are equivalent. Back to page 10 | Last page (But also see the tips and tricks page, and the competition! |