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

(links) |
(creation) |
||

Line 7: | Line 7: | ||

==<span style="font-size:20px; color: blue;">Advanced Number Theory</span>== | ==<span style="font-size:20px; color: blue;">Advanced Number Theory</span>== | ||

These are Olympiad-level theorems and properties of numbers that are routinely used on the IMO and other such competitions. | These are Olympiad-level theorems and properties of numbers that are routinely used on the IMO and other such competitions. | ||

− | < | + | ===Jensen's Inequality=== |

+ | For a convex function <math>f(x)</math> and real numbers <math>a_1,a_2,a_3,a_4\ldots,a_n</math> and <math>x_1,x_2,x_3,x_4\ldots,x_n</math>, the following holds: | ||

+ | <cmath>\sum_{i=1}^{n}a_i\cdot f(x_i)\ge f(\sum_{i=1}^{n}a_i\cdot x_i)</cmath> | ||

+ | |||

+ | ===Holder's Inequality=== | ||

+ | For positive real numbers <math>a_{i_{j}}, 1\le i\le m, 1\le j\le n be</math>, the following holds: | ||

+ | |||

+ | <cmath>\prod_{i=1}^{m}\left(\sum_{j=1}^{n}a_{i_{j}}\right)\ge\left(\sum_{j=1}^{n}\sqrt[m]{\prod_{i=1}^{m}a_{i_{j}}}\right)^{m}</cmath> | ||

+ | <!-- okay, I can't think of more. can you? --> | ||

[[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 11:07, 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: Back to page 10 | Last page (But also see the tips and tricks page, and the competition! |