# Difference between revisions of "Superagh's Olympiad Notes"

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If <math>x \ge y</math>, then<cmath>pm_x(a_1, a_2, \cdots , a_n) \ge pm_y(a_1, a_2, \cdots , a_n).</cmath> | If <math>x \ge y</math>, then<cmath>pm_x(a_1, a_2, \cdots , a_n) \ge pm_y(a_1, a_2, \cdots , a_n).</cmath> | ||

− | Power mean (weighted) | + | ====Power mean (weighted)==== |

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>, | 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>, | ||

## Revision as of 22:28, 24 June 2020

## Contents

## Introduction

SINCE MY COMPUTER WON'T LOAD THIS FOR SOME REASON, I'LL BE UPDATING THIS AS I GO THOUGH :)

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 , where . Define the as:where , and:where .

If , then

#### Power mean (weighted)

Statement: Let be positive real numbers. Let be positive real numbers ("weights") such that . For any ,

if ,

.

if ,

.

If , then . Equality occurs if and only if all the are equal.

#### Cauchy-Swartz Inequality

Let there be two sets of integers, and , such that is a positive integer, where all members of the sequences are real, then we have:Equality holds if for all , where , , or for all , where , ., or we have some constant such that for all .

#### Bernoulli's Inequality

Given that , are real numbers such that and , we have:

#### Rearrangement Inequality

Given thatandWe have:is greater than any other pairings' sum.

#### Holder's Inequality

If , , , are nonnegative real numbers and are nonnegative reals with sum of , then:This is a generalization of the Cauchy Swartz Inequality.