Difference between revisions of "User:Temperal/The Problem Solver's Resource11"
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==<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
Holder's InequalityFor positive real numbers
|
and real numbers 
and 
, the following holds:
![\[\sum_{i=1}^{n}a_i\cdot f(x_i)\ge f(\sum_{i=1}^{n}a_i\cdot x_i)\]](http://latex.artofproblemsolving.com/7/d/0/7d0d8c68decf5050eb33d8c9aa34c305cf7d4700.png)
, the following holds:
![\[\prod_{i=1}^{m}\left(\sum_{j=1}^{n}a_{i_{j}}\right)\ge\left(\sum_{j=1}^{n}\sqrt[m]{\prod_{i=1}^{m}a_{i_{j}}}\right)^{m}\]](http://latex.artofproblemsolving.com/2/7/b/27ba8cce0f09b6257bd2431f97bb6a505ff53af6.png)
