Difference between revisions of "User:Superagh"
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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: 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>. | 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>. | ||
Revision as of 11:54, 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)
Statement: Given positive integers, 
, and 
has a positive sum, and integer 
, define 
to be the following expression: ![\[(w_1a_1^x+w_2a_2^x \cdots w_na_n^x)^{\frac{1}{x}}\]](http://latex.artofproblemsolving.com/1/1/0/110b9afbe374a33c371f753048b59b5fd2cffe97.png)
When .
