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

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===Holder's Inequality===
 
===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:
 
For positive real numbers <math>a_{i_{j}}, 1\le i\le m, 1\le j\le n be</math>, the following holds:
 +
<math>
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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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===Muirhead's Inequality===
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For a sequence </math>A<math> that majorizes a sequence </math>B<math>, then given a set of positive integers </math>x_1,x_2,\ldots,x_n<math>, the following holds:
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<cmath>\sum_{sym} {x_1}^{a_1}{x_2}^{a_2}\ldots {x_n}^{a_n}\geq \sum_{sym} {x_1}^{b_1}{x_2}^{b_2}\cdots {x_n}^{b_n}</cmath>
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===Rearrangement Inequality===
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For any multi sets </math>{a_1,a_2,a_3\ldots,a_n}<math> and </math>{b_1,b_2,b_3\ldots,b_n}<math>, </math>a_1b_1+a_2b_2+\ldots+a_nb_n<math> is maximized when </math>a_k<math> is greater than or equal to exactly </math>i<math> of the other members of </math>A<math>, then </math>b_k<math> is also greater than or equal to exactly </math>i<math> of the other members of </math>B<math>.
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===Newton's Inequality===
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For non-negative real numbers </math>x_1,x_2,x_3\ldots,x_n<math> and </math>0 < k < n<math> the following holds:
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<cmath>d_k^2 \ge d_{k-1}d_{k+1}</cmath>,
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with equality exactly iff all </math>x_i<math> are equivalent.
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===Mauclarin's Inequality===
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For non-negative real numbers </math>x_1,x_2,x_3 \ldots, x_n<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>  
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<cmath>x_1 \ge \sqrt[2]{x_2} \ge \sqrt[3]{x_3}\ldots \ge \sqrt[n]{x_n}</cmath>
<!-- okay, I can't think of more. can you? -->
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 +
with equality iff all </math>x_i$ 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 11:22, 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: $<cmath>\prod_{i=1}^{m}\left(\sum_{j=1}^{n}a_{i_{j}}\right)\ge\left(\sum_{j=1}^{n}\sqrt[m]{\prod_{i=1}^{m}a_{i_{j}}}\right)^{m}</cmath>  ===Muirhead's Inequality=== For a sequence$A$that majorizes a sequence$B$, then given a set of positive integers$x_1,x_2,\ldots,x_n$, the following holds:

<cmath>\sum_{sym} {x_1}^{a_1}{x_2}^{a_2}\ldots {x_n}^{a_n}\geq \sum_{sym} {x_1}^{b_1}{x_2}^{b_2}\cdots {x_n}^{b_n}</cmath> ===Rearrangement Inequality=== For any multi sets$ (Error compiling LaTeX. ! Missing $ inserted.){a_1,a_2,a_3\ldots,a_n}$and${b_1,b_2,b_3\ldots,b_n}$,$a_1b_1+a_2b_2+\ldots+a_nb_n$is maximized when$a_k$is greater than or equal to exactly$i$of the other members of$A$, then$b_k$is also greater than or equal to exactly$i$of the other members of$B$. ===Newton's Inequality=== For non-negative real numbers$x_1,x_2,x_3\ldots,x_n$and$0 < k < n$the following holds:

<cmath>d_k^2 \ge d_{k-1}d_{k+1}</cmath>,

with equality exactly iff all$ (Error compiling LaTeX. ! Missing $ inserted.)x_i$are equivalent.  ===Mauclarin's Inequality=== For non-negative real numbers$x_1,x_2,x_3 \ldots, x_n$, the following holds:

<cmath>x_1 \ge \sqrt[2]{x_2} \ge \sqrt[3]{x_3}\ldots \ge \sqrt[n]{x_n}</cmath>

with equality iff all$ (Error compiling LaTeX. ! Missing $ inserted.)x_i$ are equivalent. Back to page 10 | Last page (But also see the tips and tricks page, and the competition!



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