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 <cmath>\sum_{sym} {x_1}^{a_1}{x_2}^{a_2}\ldots {x_n}^{a_n}\geq \sum_{sym} {x_1}^{b_1}{x_2}^{b_2}\cdots {x_n}^{b_n}</cmath>
===Rearrangement Inequality===
For any multi sets$ (Error compiling LaTeX. Unknown error_msg){a_1,a_2,a_3\ldots,a_n} <cmath>d_k^2 \ge d_{k-1}d_{k+1}</cmath>, with equality exactly iff all$ (Error compiling LaTeX. Unknown error_msg)x_i <cmath>x_1 \ge \sqrt[2]{x_2} \ge \sqrt[3]{x_3}\ldots \ge \sqrt[n]{x_n}</cmath> with equality iff all$ (Error compiling LaTeX. Unknown error_msg)x_i$ are equivalent. Back to page 10 | Last page (But also see the tips and tricks page, and the competition! |
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:
![$<cmath>\prod_{i=1}^{m}\left(\sum_{j=1}^{n}a_{i_{j}}\right)\ge\left(\sum_{j=1}^{n}\sqrt[m]{\prod_{i=1}^{m}a_{i_{j}}}\right)^{m}</cmath> ===Muirhead's Inequality=== For a sequence$](http://latex.artofproblemsolving.com/8/0/d/80dc2bc967fb86eb0272ff48b67739234178263b.png)
A
B
x_1,x_2,\ldots,x_n
{b_1,b_2,b_3\ldots,b_n}
a_1b_1+a_2b_2+\ldots+a_nb_n
a_k
i
A
b_k
i
x_1,x_2,x_3\ldots,x_n
x_1,x_2,x_3 \ldots, x_n
