User:Temperal/The Problem Solver's Resource2

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 , , with equality iff 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}S_1=S_2=S_3...=S_{k-1}=S_k$. ===Cauchy-Schwarz inequality=== For any real numbers$ (Error compiling LaTeX. Unknown error_msg)a_1,a_2,...,a_nb_1,b_2,...,b_n\displaystyle(\sum a_i^2)(\sum b_i^2) \ge (\sum a_ib_i)^2$====Cauchy-Schwarz variation==== For any real numbers$ (Error compiling LaTeX. Unknown error_msg)a_1,a_2,...,a_nb_1,b_2,...,b_n\displaystyle\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