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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 $x<y$ .

I $m_2$ 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 $a|c$ .

Power Mean Inequality

For a real number $k$ and positive real numbers $a_1, a_2, ..., a_n$ , the $k$ th power mean of the $a_i$ is

$M(k) = \left( \frac{\sum_{i=1}^n a_{i}^k}{n} \right) ^ {\frac{1}{k}}$ when $k \neq 0$ and is given by the geometric mean]] of the $a_i$ when $k = 0$ .

Diverging-Converging Theorem

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

Errata

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

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@@ Line 45: / Line 45: @@
 ===Fermat-Euler Identitity===
-If <math>gcd(a,m)=1</math>, then <math>a^{\phi{m}}\equiv1\pmod{m}</math>, where <math>\phi{m}</math> is the number of relitvely prime  numbers lower than <math>m</math>.
+If <math>gcd(a,m)=1</math>, then <math>a^{\phi{m}}\equiv1\pmod{m}</math>, where <math>\phi{m}</math> is the number of relatively prime  numbers lower than <math>m</math>.
 ===Gauss's Theorem===

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