# 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> | ||

+ | <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 </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: | ||

+ | |||

+ | <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 </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>. | ||

+ | ===Newton's Inequality=== | ||

+ | 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: | ||

+ | |||

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

+ | |||

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

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

+ | For non-negative real numbers </math>x_1,x_2,x_3 \ldots, x_n<math>, the following holds: | ||

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

− | < | + | |

+ | 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

## 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: ABx_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}{b_1,b_2,b_3\ldots,b_n}a_1b_1+a_2b_2+\ldots+a_nb_na_kiAb_kiBx_1,x_2,x_3\ldots,x_n0 < 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_ix_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! |