User:Superagh

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 $r \in \mathbb{R}$ ,