Difference between revisions of "User:Superagh"
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====Power mean (weighted)==== | ====Power mean (weighted)==== | ||
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| + | Statement: Let <math>a_1, a_2, a_3, . . . a_n</math> be positive real numbers. Let <math>w_1, w_2, w_3, . . . w_n</math> be positive real numbers ("weights") such that <math>w_1+w_2+w_3+ . . . w_n=1</math>. For any <math>r \in \mathbb{R}</math>, | ||
| + | |||
| + | if <math>r=0</math>, | ||
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| + | <math>P(r)=a_1^w_1a_2^2_wa_3^w_3 . . . a_n^w_n</math>. | ||
| + | |||
| + | if <math>r \neq 0</math>, | ||
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| + | <math>P(r)=(w_1a_1^r+w_2a_2^r+w_3a_3^r . . . +w_na_n^r)^{\frac{1}{r}}</math>. | ||
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| + | If <math>r>s</math>, then <math>P(r) \geq P(s)</math>. Equality occurs if and only if all the <math>a_i</math> are equal. | ||
==Combinatorics== | ==Combinatorics== | ||
Revision as of 15:35, 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: Let 
be positive real numbers. Let 
be positive real numbers ("weights") such that 
. For any 
,
if 
,
$P(r)=a_1^w_1a_2^2_wa_3^w_3 . . . a_n^w_n$ (Error compiling LaTeX. Unknown error_msg).
if 
,
.
If 
, then 
. Equality occurs if and only if all the 
are equal.
