Difference between revisions of "User superagh olympiad notes"
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− | Introduction | + | ==Introduction== |
SINCE MY COMPUTER WON'T LOAD THIS FOR SOME REASON, I'LL BE UPDATING THIS AS I GO THOUGH :) | 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) | 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 | + | ==Algebra== |
Problems worth noting/reviewing | Problems worth noting/reviewing | ||
I'll leave this empty for now, I want to start on HARD stuff yeah! | I'll leave this empty for now, I want to start on HARD stuff yeah! | ||
− | Inequalities | + | ===Inequalities=== |
We shall begin with INEQUALITIES! They should be fun enough. I should probably begin with some theorems. | We shall begin with INEQUALITIES! They should be fun enough. I should probably begin with some theorems. | ||
− | Power mean (special case) | + | ====Power mean (special case)==== |
Statement: Given that <math>a_1, a_2, a_3, ... a_n > 0</math>, <math>a_{i} \in \mathbb{R}</math> where <math>1 \le i \le n</math>. Define the <math>pm_x(a_1, a_2, \cdots , a_n)</math> as:<cmath>(\frac{a_1^x+a_2^x+\cdots+a_n^x}{n})^{\frac{1}{x}},</cmath>where <math>x\neq0</math>, and:<cmath>\sqrt[n]{a_{1}a_{2}a_{3} \cdots a_{n}}.</cmath>where <math>x=0</math>. | Statement: Given that <math>a_1, a_2, a_3, ... a_n > 0</math>, <math>a_{i} \in \mathbb{R}</math> where <math>1 \le i \le n</math>. Define the <math>pm_x(a_1, a_2, \cdots , a_n)</math> as:<cmath>(\frac{a_1^x+a_2^x+\cdots+a_n^x}{n})^{\frac{1}{x}},</cmath>where <math>x\neq0</math>, and:<cmath>\sqrt[n]{a_{1}a_{2}a_{3} \cdots a_{n}}.</cmath>where <math>x=0</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. | 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. | ||
− | Cauchy-Swartz Inequality | + | ====Cauchy-Swartz Inequality==== |
Let there be two sets of integers, <math>a_1, a_2, \cdots a_n</math> and <math>b_1, b_2, \cdots b_n</math>, such that <math>n</math> is a positive integer, where all members of the sequences are real, then we have:<cmath>(a_1^2+a_2^2+\cdots +a_n^2)(b_1^2+b_2^2+ \cdots +b_n^2)\ge (a_1b_1 + a_2b_2 + \cdots +a_nb_n)^2.</cmath>Equality holds if for all <math>a_i</math>, where <math>1\le i \le n</math>, <math>a_i=0</math>, or for all <math>b_i</math>, where <math>1\le i \le n</math>, <math>b_i=0</math>., or we have some constant <math>k</math> such that <math>b_i=ka_i</math> for all <math>i</math>. | Let there be two sets of integers, <math>a_1, a_2, \cdots a_n</math> and <math>b_1, b_2, \cdots b_n</math>, such that <math>n</math> is a positive integer, where all members of the sequences are real, then we have:<cmath>(a_1^2+a_2^2+\cdots +a_n^2)(b_1^2+b_2^2+ \cdots +b_n^2)\ge (a_1b_1 + a_2b_2 + \cdots +a_nb_n)^2.</cmath>Equality holds if for all <math>a_i</math>, where <math>1\le i \le n</math>, <math>a_i=0</math>, or for all <math>b_i</math>, where <math>1\le i \le n</math>, <math>b_i=0</math>., or we have some constant <math>k</math> such that <math>b_i=ka_i</math> for all <math>i</math>. | ||
− | 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. | ||
− | Holder's Inequality | + | ====Holder's Inequality==== |
If <math>a_1, a_2, \cdots, a_n</math>, <math>b_1, b_2, \cdots, b_n</math>, <math>\cdots</math>, <math>z_1, z_2, \cdots, z_n</math> are nonnegative real numbers and <math>\lambda_a, \lambda_b, \cdots, \lambda_z</math> are nonnegative reals with sum of <math>1</math>, then:<cmath>a_1^{\lambda_a}b_1^{\lambda_b} \cdots z_1^{\lambda_z} + \cdots + a_n^{\lambda_a} b_n^{\lambda_b} \cdots z_n^{\lambda_z} \le (a_1 + \cdots + a_n)^{\lambda_a} (b_1 + \cdots + b_n)^{\lambda_b} \cdots (z_1 + \cdots + z_n)^{\lambda_z} .</cmath>This is a generalization of the Cauchy Swartz Inequality. | If <math>a_1, a_2, \cdots, a_n</math>, <math>b_1, b_2, \cdots, b_n</math>, <math>\cdots</math>, <math>z_1, z_2, \cdots, z_n</math> are nonnegative real numbers and <math>\lambda_a, \lambda_b, \cdots, \lambda_z</math> are nonnegative reals with sum of <math>1</math>, then:<cmath>a_1^{\lambda_a}b_1^{\lambda_b} \cdots z_1^{\lambda_z} + \cdots + a_n^{\lambda_a} b_n^{\lambda_b} \cdots z_n^{\lambda_z} \le (a_1 + \cdots + a_n)^{\lambda_a} (b_1 + \cdots + b_n)^{\lambda_b} \cdots (z_1 + \cdots + z_n)^{\lambda_z} .</cmath>This is a generalization of the Cauchy Swartz Inequality. | ||
− | Combinatorics | + | ==Combinatorics== |
− | Number Theory | + | ==Number Theory== |
− | Geometry | + | ==Geometry== |
Revision as of 19:12, 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 thatandWe 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.