MATHSHELPER FOR STUDENTS
Firstly I would like to recommend a couple of books that are superb at helping you improve at olympiad type problems.
A Mathematical Olympiad Primer by Geoff Smith.
The Mathematical Olympiad Handbook by A. Gardner.
Initially these problems can seem impossible, but with perseverance you can improve at them. I did dreadfully in the BMO1 when I took it at school, but I can now do the majority of the questions they throw at you. (Alas I still find BMO2 beyond my abilities. I would wager that you could give me the rest of my life to solve question one from the 2010 paper and I would still not be able to do it!) Everyone gets stuck at some point or other on a problem. It's how you approach these difficulties that matters.
It can be argued that you only really learn anything from the times you get stuck.
Secondly you can find the British Maths Olympiad here with all the past papers, but without answers unfortunately. The UKMT provides some cheap booklets with BMO problems and solutions from previous years. I can also recommend the UKMT yearbooks which provide all the Olympiad probems and answers with discussions.
ALL BMO PAPERS IN ONE DOCUMENT
In this section
Why did the chicken cross the Mobius Strip?
To get to the same side
What's purple and commutes?
An Abelian grape