|
|
| (19 intermediate revisions by 2 users not shown) |
| Line 1: |
Line 1: |
| − | ==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 <math>a_1, a_2, a_3, ... a_n > 0</math>, <math>a_{i} \in \mathbb{R}</math> where <math>1 \le i \le n</math>. Define the <math>pm_x(a_1, a_2, \cdots , a_n)</math> as: <cmath>(\frac{a_1^x+a_2^x+\cdots+a_n^x}{n})^{\frac{1}{x}},</cmath> where <math>x\neq0</math>, and: <cmath>\sqrt[n]{a_{1}a_{2}a_{3} \cdots a_{n}}.</cmath> where <math>x=0</math>.
| |
| − |
| |
| − | 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)=====
| |
| − | Statement: Given positive integers, <math>a_1, a_2, a_3 \cdots a_n</math>, and <math>w_1, w_2 \cdots w_n</math> has a positive sum, and integer <math>x</math>, define <math>pm_x(a_1, a_2, \cdots a_n)</math> to be the following expression: <cmath>(w_1a_1^x+w_2a_2^x \cdots w_na_n^x)^{\frac{1}{x}}</cmath> When <math>x\neq0</math>.
| |
| − |
| |
| − | ==Combinatorics==
| |
| − |
| |
| − | ==Number Theory==
| |
| − |
| |
| − | ==Geometry==
| |
