# User:Temperal/The Problem Solver's Resource11

The Problem Solver's Resource
 Introduction | Other Tips and Tricks | Methods of Proof | You are currently viewing page 11.

## Advanced Number Theory

These are Olympiad-level theorems and properties of numbers that are routinely used on the IMO and other such competitions.

### Jensen's Inequality

For a convex function $f(x)$ and real numbers $a_1,a_2,a_3,a_4\ldots,a_n$ and $x_1,x_2,x_3,x_4\ldots,x_n$, the following holds:

$$\sum_{i=1}^{n}a_i\cdot f(x_i)\ge f(\sum_{i=1}^{n}a_i\cdot x_i)$$

### Holder's Inequality

For positive real numbers $a_{i_{j}}, 1\le i\le m, 1\le j\le n be$, the following holds: $\prod_{i=1}^{m}\left(\sum_{j=1}^{n}a_{i_{j}}\right)\ge\left(\sum_{j=1}^{n}\sqrt[m]{\prod_{i=1}^{m}a_{i_{j}}}\right)^{m} ===Muirhead's Inequality=== For a sequence$A$that majorizes a sequence$B$, then given a set of positive integers$x_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 $inserted.){a_1,a_2,a_3\ldots,a_n}$and${b_1,b_2,b_3\ldots,b_n}$,$a_1b_1+a_2b_2+\ldots+a_nb_n$is maximized when$a_k$is greater than or equal to exactly$i$of the other members of$A$, then$b_k$is also greater than or equal to exactly$i$of the other members of$B$. ===Newton's Inequality=== For non-negative real numbers$x_1,x_2,x_3\ldots,x_n$and$0 < k < n$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_i$are equivalent. ===Mauclarin's Inequality=== For non-negative real numbers$x_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!