User:Superagh

Revision as of 09:25, 24 June 2020 by Superagh (talk | contribs) (Power mean)

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.

Theorems worth noting

Power mean

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 $x\gey$ (Error compiling LaTeX. Unknown error_msg), then

\[pm_x(a_1, a_2, \cdots , a_n)\gepm_y(a_1, a_2, \cdots , a_n).\] (Error compiling LaTeX. Unknown error_msg)

Combinatorics

Number Theory

Geometry