Difference between revisions of "User:Temperal/The Problem Solver's Resource2"

Revision as of 15:52, 29 September 2007

The Problem Solver's Resource

Introduction \| Other Tips and Tricks \| Methods of Proof \| You are currently viewing page 2.

Simple Number Theory

This is a collection of essential AIME-level number theory theorems and other tidbits.

Trivial Inequality

For any real $x$ , $x^2\ge 0$ , with equality iff $x=0$ .

Arithmetic Mean/Geometric Mean Inequality

For any set of real numbers $S$ , $\frac{S_1+S_2+S_3....+S_{k-1}+S_k}{k}\ge \sqrt[k]{S_1\cdot S_2 \cdot S_3....\cdot S_{k-1}\cdot S_k}$ with equality iff $S_1=S_2=S_3...=S_{k-1}=S_k$ .

Cauchy-Schwarz inequality

For any real numbers $a_1,a_2,...,a_n$ and $b_1,b_2,...,b_n$ , the following holds:

$(\sum a_i^2)(\sum b_i^2) \ge (\sum a_ib_i)^2$

Cauchy-Schwarz variation

For any real numbers $a_1,a_2,...,a_n$ and positive real numbers $b_1,b_2,...,b_n$ , the following holds:

$\sum\left({{a_i^2}\over{b_i}}\right) \ge {{\sum a_i^2}\over{\sum b_i}}$ .

Back to page 1 | Continue to page 3

Revision as of 15:52, 29 September 2007 (view source) Temperal (talk \| contribs) ← Older edit		Revision as of 15:52, 29 September 2007 (view source) Temperal (talk \| contribs) (→‎Cauchy-Schwarz variation) Newer edit →
Line 27:		Line 27:


−	[[User:Temperal/The Problem Solver's Resource1\|Back to page 1]] \| [[User:Temperal/The Problem Solver's ~~Resource2~~\|Continue to page 3]]	+	[[User:Temperal/The Problem Solver's Resource1\|Back to page 1]] \| [[User:Temperal/The Problem Solver's Resource3\|Continue to page 3]]
	\|}<br /><br />		\|}<br /><br />

Page

Toolbox

Search