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.
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===Jensen's Inequality===
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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:
  
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<cmath>\sum_{i=1}^{n}a_i\cdot f(x_i)\ge f(\sum_{i=1}^{n}a_i\cdot x_i)</cmath>
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===Holder's Inequality===
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For positive real numbers <math>a_{i_{j}}, 1\le i\le m, 1\le j\le n be</math>, the following holds:
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<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>
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[[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]]!
 
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Revision as of 11:07, 13 October 2007



The Problem Solver's Resource
Introduction | Other Tips and Tricks | Methods of Proof | You are currently viewing page 11.

Advanced Number Theory

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 $f(x)$ and real numbers $a_1,a_2,a_3,a_4\ldots,a_n$ and $x_1,x_2,x_3,x_4\ldots,x_n$, the following holds:

\[\sum_{i=1}^{n}a_i\cdot f(x_i)\ge f(\sum_{i=1}^{n}a_i\cdot x_i)\]

Holder's Inequality

For positive real numbers $a_{i_{j}}, 1\le i\le m, 1\le j\le n be$, the following holds:

\[\prod_{i=1}^{m}\left(\sum_{j=1}^{n}a_{i_{j}}\right)\ge\left(\sum_{j=1}^{n}\sqrt[m]{\prod_{i=1}^{m}a_{i_{j}}}\right)^{m}\] Back to page 10 | Last page (But also see the tips and tricks page, and the competition!



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