# Difference between revisions of "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.

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:

$$\prod_{i=1}^{m}\left(\sum_{j=1}^{n}a_{i_{j}}\right)\ge\left(\sum_{j=1}^{n}\sqrt[m]{\prod_{i=1}^{m}a_{i_{j}}}\right)^{m}$$ Back to page 10 | Last page (But also see the tips and tricks page, and the competition!