Difference between revisions of "User:Temperal/The Problem Solver's Resource8"
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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>. | ||
+ | |||
+ | ===Power Mean Inequality=== | ||
+ | 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 | ||
+ | |||
+ | <math>M(k) = \left( \frac{\sum_{i=1}^n a_{i}^k}{n} \right) ^ {\frac{1}{k}}</math> | ||
+ | when <math>k \neq 0</math> and is given by the geometric mean]] of the | ||
+ | <math>a_i</math> when <math>k = 0</math>. | ||
===Errata=== | ===Errata=== |
Revision as of 22:11, 5 October 2007
Intermediate Number TheoryThese 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 InequalityTake a set of functions Note that
I Chebyshev's InequalityGiven real numbers %{\frac{\sum a_ib_i}{n}} \ge {\frac{\sum a_i}{n}}{\frac{\sum b_i}{n}}%. Minkowsky's InequalityGiven real numbers
Nesbitt's InequalityFor all positive real numbers
Schur's inequalityGiven positive real numbers
Fermat-Euler IdentitityIf Gauss's TheoremIf Power Mean InequalityFor a real number
ErrataAll quadratic resiues are |