User:Temperal/The Problem Solver's Resource11
Advanced Number TheoryThese are Olympiad-level theorems and properties of numbers that are routinely used on the IMO and other such competitions. Jensen's InequalityFor a convex function
Holder's InequalityFor positive real numbers
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and real numbers 
and 
, the following holds:
![\[\sum_{i=1}^{n}a_i\cdot f(x_i)\ge f(\sum_{i=1}^{n}a_i\cdot x_i)\]](http://latex.artofproblemsolving.com/7/d/0/7d0d8c68decf5050eb33d8c9aa34c305cf7d4700.png)
, the following holds:
![\[\prod_{i=1}^{m}\left(\sum_{j=1}^{n}a_{i_{j}}\right)\ge\left(\sum_{j=1}^{n}\sqrt[m]{\prod_{i=1}^{m}a_{i_{j}}}\right)^{m}\]](http://latex.artofproblemsolving.com/2/7/b/27ba8cce0f09b6257bd2431f97bb6a505ff53af6.png)
