(Cross-posted in MESE 8173.)
I want to start to do mathematical Olympiad type questions but have absolutely no knowledge on how to solve these apart from my school curriculum. I'm $16$ but know maths up to the $18$ year old level. I think I will start learning the theory of the topics (Elementary Number Theory, Combinatorics, Euclidean Plane Geometry) then going on to trying the questions, but I need help in knowing what books to use to learn the theory. I have seen several list (such as http://www.artofproblemsolving.com/wiki/index.php/Olympiad_books) but does anyone know which ones are the best for my level of knowledge.
P.S. I live in UK if that matters.
I understand your reluctance to focus solely on contest books. Often some of the "tricks" used are really applications of some more general theory in disguise. Greater clarity is sometimes brought by studying math at a higher level or at least in a more orderly way (from books where the exposition is directed at a greater purpose).
The books in this series tend to be good and are aimed at a high school audience. I particularly like the two by Niven.
You could also study some number theory (for example from the book by Stark). That tends to be closely related to a lot of contest material.
In parallel, you shouldn't hesitate to study math at university level now - particularly rigorous calculus (such as in Spivak's book) and algebra (e.g. from Artin's book). When I was in high school, my contest-type ability went up a lot in the space of a year when I started studying analysis and abstract algebra on my own.
Plane geometry needs to be studied in its own right for contests, however. One doesn't get that knowledge from typical university-level reading. Geometry Revisited, which appears in the series I linked to, is excellent in that respect. So too are Yaglom's books.