Difference between revisions of "Superagh's Olympiad Notes"
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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) | 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) | ||
+ | |||
+ | NEVERYMIND I"M DOING THIS ON MY BLOG SINCE IT"LL LOAD | ||
==Algebra== | ==Algebra== | ||
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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: 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>, | 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>, | ||
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==Combinatorics== | ==Combinatorics== | ||
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+ | ok look bro | ||
+ | |||
+ | <math>a^2+b^2=c^2</math> | ||
+ | |||
+ | lol | ||
+ | |||
+ | <math>\phi{n} \equiv 1 \pmod{asdf}</math>. | ||
+ | |||
==Number Theory== | ==Number Theory== | ||
==Geometry== | ==Geometry== | ||
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+ | hahaha | ||
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+ | geometry sucks | ||
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+ | ~Lcz 5:02 PM CST, 6/25/2020 |
Latest revision as of 13:59, 22 July 2020
Contents
Introduction
SINCE MY COMPUTER WON'T LOAD THIS FOR SOME REASON, I'LL BE UPDATING THIS AS I GO THOUGH :)
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)
NEVERYMIND I"M DOING THIS ON MY BLOG SINCE IT"LL LOAD
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:
where
, and:
where
.
If , then
Power mean (weighted)
Statement: Let be positive real numbers. Let
be positive real numbers ("weights") such that
. For any
,
if ,
.
if ,
.
If , then
. Equality occurs if and only if all the
are equal.
Cauchy-Swartz Inequality
Let there be two sets of integers, and
, such that
is a positive integer, where all members of the sequences are real, then we have:
Equality holds if for all
, where
,
, or for all
, where
,
., or we have some constant
such that
for all
.
Bernoulli's Inequality
Given that ,
are real numbers such that
and
, we have:
Rearrangement Inequality
Given thatand
We have:
is greater than any other pairings' sum.
Holder's Inequality
If ,
,
,
are nonnegative real numbers and
are nonnegative reals with sum of
, then:
This is a generalization of the Cauchy Swartz Inequality.
Combinatorics
ok look bro
lol
.
Number Theory
Geometry
hahaha
geometry sucks
~Lcz 5:02 PM CST, 6/25/2020