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Difference between revisions of "User:Temperal/The Problem Solver's Resource8"

(→‎General Mean Inequality: inequality)
(I think the Power mean inequality is the same thing as the general mean inequality (at least, their descriptions here are))
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===Gauss's Theorem===
 
===Gauss's Theorem===
 
If <math>a|bc</math> and <math>(a,b) = 1</math>, then <math>a|c</math>.
 
If <math>a|bc</math> and <math>(a,b) = 1</math>, then <math>a|c</math>.
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===Power Mean Inequality===
 
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For a real number <math>k</math> and positive real numbers <math>a_1, a_2, ..., a_n</math>, the <math>k</math>th power mean of the <math>a_i</math> is
 
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<math>M(k) = \left( \frac{\sum_{i=1}^n a_{i}^k}{n} \right) ^ {\frac{1}{k}}</math>
 
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when <math>k \neq 0</math> and is given by the geometric mean]] of the
 
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<math>a_i</math> when <math>k = 0</math>.
 
  
 
===Diverging-Converging Theorem===
 
===Diverging-Converging Theorem===

Revision as of 17:40, 9 October 2007



The Problem Solver's Resource
Introduction | Other Tips and Tricks | Methods of Proof | You are currently viewing page 8.

Intermediate Number Theory

These are more complex number theory theorems that may turn up on the USAMO or Pre-Olympiad tests. This will also cover diverging and converging series, and other such calculus-related topics.

General Mean Inequality

Take a set of functions $m_j(a) = \left({\frac{\sum a_i^j}{n}}\right)^{1/j}$.

Note that $m_0$ does not exist. The geometric mean is $m_0 = \lim_{k \to 0} m_k$. For non-negative real numbers $a_1,a_2,\ldots,a_n$, the following holds:

$m_x \le m_y$ for reals $x<y$.

, if $m_2$ is the quadratic mean, $m_1$ is the arithmetic mean, $m_0$ the geometric mean, and $m_{-1}$ the harmonic mean.

Chebyshev's Inequality

Given real numbers $a_1 \ge a_2 \ge ... \ge a_n \ge 0$ and $b_1 \ge b_2 \ge ... \ge b_n$, we have

${\frac{\sum a_ib_i}{n}} \ge {\frac{\sum a_i}{n}}{\frac{\sum b_i}{n}}$.

Minkowsky's Inequality

Given real numbers $a_1,a_2,...,a_n$ and $b_1,b_2,\ldots,b_n$, the following holds:

$\sqrt{\sum a_i^2} + \sqrt{\sum b_i^2} \ge \sqrt{\sum (a_i+b_i)^2}$

Nesbitt's Inequality

For all positive real numbers $a$, $b$ and $c$, the following holds:

${\frac{a}{b+c}} + {\frac{b}{c+a}} + {\frac{c}{a+b}} \ge {\frac{3}{2}}$.

Schur's inequality

Given positive real numbers $a,b,c$ and real $r$, the following holds:

$a^r(a-b)(a-c)+b^r(b-a)(b-c)+c^r(c-a)(c-b)\ge 0$.

Fermat-Euler Identitity

If $gcd(a,m)=1$, then $a^{\phi{m}}\equiv1\pmod{m}$, where $\phi{m}$ is the number of relatively prime numbers lower than $m$.

Gauss's Theorem

If $a|bc$ and $(a,b) = 1$, then $a|c$.

Diverging-Converging Theorem

A series $\sum_{i=0}^{\infty}S_i$ converges iff $\lim S_i=0$.

Errata

All quadratic residues are $0$ or $1\pmod{4}$and $0$, $1$, or $4$ $\pmod{8}$.


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