User:Temperal/The Problem Solver's Resource8
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. Power Mean InequalityTake a set of functions . Note that does not exist. The geometric mean is . For non-negative real numbers , the following holds: for reals . , if is the quadratic mean, is the arithmetic mean, the geometric mean, and the harmonic mean. Chebyshev's InequalityGiven real numbers and , we have . Minkowsky's InequalityGiven real numbers and , the following holds:
Nesbitt's InequalityFor all positive real numbers , and , the following holds: . Schur's inequalityGiven positive real numbers and real , the following holds: . Fermat-Euler IdentitityIf , then , where is the number of relatively prime numbers lower than . Gauss's TheoremIf and , then . Diverging-Converging TheoremA series converges iff . ErrataAll quadratic residues are or and , , or . |