User:Temperal/The Problem Solver's Resource2

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Revision as of 15:52, 29 September 2007 by Temperal (talk | contribs) (Cauchy-Schwarz variation)

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:

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

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