Difference between revisions of "Superagh's Olympiad Notes"
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====Bernoulli's Inequality==== | ====Bernoulli's Inequality==== | ||
Given that <math>n</math>, <math>x</math> are real numbers such that <math>n\ge 0</math> and <math>x \ge -1</math>, we have:<cmath>(1+x)^n \ge 1+nx.</cmath> | Given that <math>n</math>, <math>x</math> are real numbers such that <math>n\ge 0</math> and <math>x \ge -1</math>, we have:<cmath>(1+x)^n \ge 1+nx.</cmath> | ||
− | Rearrangement Inequality | + | ====Rearrangement Inequality==== |
Given that<cmath>x_1 \ge x_2 \ge x_3 \cdots x_n</cmath>and<cmath>y_1 \ge y_2 \ge y_3 \cdots y_n.</cmath>We have:<cmath>x_1y_1+x_2y_2 + \cdots + x_ny_n</cmath>is greater than any other pairings' sum. | Given that<cmath>x_1 \ge x_2 \ge x_3 \cdots x_n</cmath>and<cmath>y_1 \ge y_2 \ge y_3 \cdots y_n.</cmath>We have:<cmath>x_1y_1+x_2y_2 + \cdots + x_ny_n</cmath>is greater than any other pairings' sum. | ||
Revision as of 20:57, 24 June 2020
Contents
[hide]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)
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.