What are the math challenges websites? I know two websites that offer some challening puzzles for programming, Project Euler (PE has something about math, but I feel it's more about programming) and Code-Golf.
Can you recommend me some sites that have the same purpose for maths? Until now I know only The Art of Problem Solving. You can recommend me these websites in any level.
 A: Wanted to make this just as a comment but I do not have enough reputation yet.
Google code jam offers quite a number of programming challenges, but only a portion of them are focused on Mathematics.
Most of them are problems from Computer Science concerning algorithms, complexity etc. However, a number of them are purely Mathematical and may be what you are searching for.
For example, there is a question about subset sums and another on probability.
Similarly, you might want to consider visiting programming sites like Top coder and  other similar ones to pick out the Mathematics programming questions, instead of the other way round. (Programming sites with only Maths problems)
Edit: Missed out one additional programming puzzle site by stackexchange itself
A: Of course there are print sources of problems.  For example The American Mathematical Monthly column of problems and solutions.  And a dozen or so other similar publications.  (I once had a list of those, I wonder where it is...)
A: Most of these links contain competition style problems. I'm not sure if that's what you're looking for or not? The links with the notes have a lot of problems done out. You might want to ignore the solutions in the ones where it's offered.
If you just want a site that asks very simple questions, often with very intricate answers visit http://www.qbyte.org/puzzles/ I don't think it's updated anymore, but there are 160 problems that offer a lot of fun. enjoy!
Somevery good notes from the American Olympiad Training programme:
http://www.math.cmu.edu/~ploh/olympiad.shtml
Olympiad training notes from Australia:
http://school.maths.uwa.edu.au/~gregg/Olympiad/
Many past years of IMO questions with solutions:
http://sms.math.nus.edu.sg/simo/simo.aspx
See also http://nepalimath.com/olympiad.aspx
which has a lot of good resources including ideas on how to solve problems.
The New Zealand Maths Olympiad Committee Online
http://www.mathsolympiad.org.nz/category/notes/
has some fantastic notes, problems and solutions.
For example, if you click on the link there, you’ll then be able to click on “notes” by Arkadii Slinko which takes you to
http://www.mathsolympiad.org.nz/wp-content/uploads/2009/04/geometric-inequalities.pdf
At http://www.mathsolympiad.org.nz/2011/05/2011-may-problems/,
click on the word “problems” 
While some of the links at http://olympiads.win.tue.nl/imo/#archives are not active, others are, and contain some fascinating material and problems
One final site is http://www.physics.harvard.edu/academics/undergrad/problems.html It isn't really competition problems at all. but has some very interesting real life applications of mathematics woven into its problems...
A: Alexander Bogomolny's site Cut the Knot http://www.cut-the-knot.com/ has (among other things) problems and puzzles.
A: My list:


*

*http://mathschallenge.net/ Contains only math problems (no programming puzzles), with difficulty rate assigned to each problem.

*http://www.qbyte.org/puzzles/ Nick's mathematical puzzles

*http://www.math.purdue.edu/pow/ Problem of the week released by Math department of Purdue

A: A good site with online competition is :i-olymp. 
Best wishes !
A: A List of Blogs (mostly high level Mathematics). Note that these will not be problem solving for fun kind of sites, rather focus on cutting-edge research problems or problems that involve a lot of dedicated theoretical background. I am posting this list because you have specifically mentioned "any level" in your question.
The List:


*

*Problem Solver's Paradise

*Blog on Inequalities - Contains several inequalities related problems

*Annoying Precision - A blog with more focus on simplifying explanations on various mathematical topics

*Combinatorics related

*Godel's Letter and P = NP (Blog Focused on Complexity Theory, Quantum Computation and Formal Logic)

*Math less traveled

*Area777 - Mostly Topology related


Note that because I am a new user, I can only post 2 links per post. So I'll add the rest in the comments. If my rep increases, I'll come back and edit it.
EDIT: Combined my other post with this one as I now have the rep to post in a single answer :)
A: You might want to check out Math problem of the Month by Bilkent University.
http://www.fen.bilkent.edu.tr/~cvmath/Problem/problem.htm
I found that the questions are rather original and are challenging too.
Hope you enjoy it.
A: Considering the level of the AoPS site, I don't know what else are you looking for. :)
Anyway, I'm Italian and we have an "olympic" website: it's basically the same as AoPS, but with a bigger number of easy problems, so it is much better suited for beginners. Maybe you can look for something similar in your language!
