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:
Olympiad training notes from Australia:
Many past years of IMO questions with solutions:
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
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
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...