Difference between revisions of "User:Superagh"
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====Power mean (weighted)==== | ====Power mean (weighted)==== | ||
| − | + | For all positive reals <math>a_1, \ldots, a_n</math> and nonnegative reals <math>p_1, \ldots, p_n</math> with <math>p_1+p_2+\cdots + p_n=1</math>, we have | |
| − | + | <cmath>\prod_{i=1}^n a_i^{p_i} \leq \sum_{i=1}^n p_ia_i.</cmath> | |
==Combinatorics== | ==Combinatorics== | ||
Revision as of 16:26, 24 June 2020
Contents
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 
, 
where 
. Define the 
as: ![\[(\frac{a_1^x+a_2^x+\cdots+a_n^x}{n})^{\frac{1}{x}},\]](http://latex.artofproblemsolving.com/e/b/c/ebc47ff3b8f1015a599578cca800e555507e382c.png)
where 
, and: ![\[\sqrt[n]{a_{1}a_{2}a_{3} \cdots a_{n}}.\]](http://latex.artofproblemsolving.com/a/2/d/a2dd19fdaf0686ffca569aceaf5f1ec2f02916f3.png)
where 
.
If 
, then ![\[pm_x(a_1, a_2, \cdots , a_n) \ge pm_y(a_1, a_2, \cdots , a_n).\]](http://latex.artofproblemsolving.com/7/a/5/7a51f71d68c4e95c1e85575a6681d32c8ad46da0.png)
Power mean (weighted)
For all positive reals 
and nonnegative reals 
with 
, we have
![\[\prod_{i=1}^n a_i^{p_i} \leq \sum_{i=1}^n p_ia_i.\]](http://latex.artofproblemsolving.com/a/0/f/a0fee4dcbb7758643b8a8bdbf6407944a5e91da9.png)
