# 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 and real numbers and , the following holds:
## Holder's InequalityFor positive real numbers , the following holds: ABx_1,x_2,\ldots,x_n$, the following holds: <cmath>\sum_{sym} {x_1}^{a_1}{x_2}^{a_2}\ldots {x_n}^{a_n}\geq \sum_{sym} {x_1}^{b_1}{x_2}^{b_2}\cdots {x_n}^{b_n}</cmath> ===Rearrangement Inequality=== For any multi sets$ (Error compiling LaTeX. ! Missing $ inserted.){a_1,a_2,a_3\ldots,a_n}{b_1,b_2,b_3\ldots,b_n}a_1b_1+a_2b_2+\ldots+a_nb_na_kiAb_kiBx_1,x_2,x_3\ldots,x_n0 < k < n$the following holds: <cmath>d_k^2 \ge d_{k-1}d_{k+1}</cmath>, with equality exactly iff all$ (Error compiling LaTeX. ! Missing $ inserted.)x_ix_1,x_2,x_3 \ldots, x_n$, the following holds: <cmath>x_1 \ge \sqrt[2]{x_2} \ge \sqrt[3]{x_3}\ldots \ge \sqrt[n]{x_n}</cmath> with equality iff all$ (Error compiling LaTeX. ! Missing $ inserted.)x_i$ are equivalent. Back to page 10 | Last page (But also see the tips and tricks page, and the competition! |