User:Temperal/The Problem Solver's Resource11
Advanced Number Theory
These are Olympiad-level theorems and properties of numbers that are routinely used on the IMO and other such competitions.
For a convex function and real numbers and , the following holds:
For positive real numbers , the following holds:
For a sequence that majorizes a sequence , then given a set of positive integers , the following holds:
For any multi sets and , is maximized when is greater than or equal to exactly of the other members of , then is also greater than or equal to exactly of the other members of .
For non-negative real numbers and the following holds:
with equality exactly iff all are equivalent.
For non-negative real numbers , and such that , for the following holds:
with equality iff all are equivalent.