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The Problem Solver's Resource

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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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@@ Line 14: / Line 14: @@
-===Cauchy-Schwarz inequality===
+===Cauchy-Schwarz Inequality===
 For any real numbers <math>a_1,a_2,...,a_n</math> and <math>b_1,b_2,...,b_n</math>, the following holds:
@@ Line 20: / Line 20: @@
 <math>(\sum a_i^2)(\sum b_i^2) \ge (\sum a_ib_i)^2</math>
-====Cauchy-Schwarz variation====
+====Cauchy-Schwarz Variation====
 For any real numbers <math>a_1,a_2,...,a_n</math> and positive real numbers <math>b_1,b_2,...,b_n</math>, the following holds:

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