Difference between revisions of "User:Superagh"

(Power mean (weighted))
(Power mean (weighted))
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====Power mean (weighted)====
 
====Power mean (weighted)====
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Statement: Let <math>a_1, a_2, a_3, . . . a_n</math> be positive real numbers. Let <math>w_1, w_2, w_3, . . . w_n</math> be positive real numbers ("weights") such that <math>w_1+w_2+w_3+ . . . w_n=1</math>. For any <math>r \in \mathbb{R}</math>,
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if <math>r=0</math>,
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<math>P(r)=a_1^w_1a_2^2_wa_3^w_3 . . . a_n^w_n</math>.
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if <math>r \neq 0</math>,
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<math>P(r)=(w_1a_1^r+w_2a_2^r+w_3a_3^r . . . +w_na_n^r)^{\frac{1}{r}}</math>.
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If <math>r>s</math>, then <math>P(r) \geq P(s)</math>. Equality occurs if and only if all the <math>a_i</math> are equal.
  
 
==Combinatorics==
 
==Combinatorics==

Revision as of 16:35, 24 June 2020

Introduction

Ok, so inspired by master math solver Lcz, I have decided to take Oly notes (for me) online! I'll probably be yelled at even more for staring at the computer, but I know that this is for my good. (Also this thing is almost the exact same format as Lcz's :P ). (Ok, actually, a LOT of credits to Lcz)

Algebra

Problems worth noting/reviewing

I'll leave this empty for now, I want to start on HARD stuff yeah!

Inequalities

We shall begin with INEQUALITIES! They should be fun enough. I should probably begin with some theorems.

Power mean (special case)

Statement: Given that $a_1, a_2, a_3, ... a_n > 0$, $a_{i} \in \mathbb{R}$ where $1 \le i \le n$. Define the $pm_x(a_1, a_2, \cdots , a_n)$ as: \[(\frac{a_1^x+a_2^x+\cdots+a_n^x}{n})^{\frac{1}{x}},\] where $x\neq0$, and: \[\sqrt[n]{a_{1}a_{2}a_{3} \cdots a_{n}}.\] where $x=0$.

If $x \ge y$, then \[pm_x(a_1, a_2, \cdots , a_n) \ge pm_y(a_1, a_2, \cdots , a_n).\]

Power mean (weighted)

Statement: Let $a_1, a_2, a_3, . . . a_n$ be positive real numbers. Let $w_1, w_2, w_3, . . . w_n$ be positive real numbers ("weights") such that $w_1+w_2+w_3+ . . . w_n=1$. For any $r \in \mathbb{R}$,

if $r=0$,

$P(r)=a_1^w_1a_2^2_wa_3^w_3 . . . a_n^w_n$ (Error compiling LaTeX. Unknown error_msg).

if $r \neq 0$,

$P(r)=(w_1a_1^r+w_2a_2^r+w_3a_3^r . . . +w_na_n^r)^{\frac{1}{r}}$.

If $r>s$, then $P(r) \geq P(s)$. Equality occurs if and only if all the $a_i$ are equal.

Combinatorics

Number Theory

Geometry