(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.

• There's a book by Terence Tao that sounds like the kind of thing you want: amazon.com/Solving-Mathematical-Problems-Personal-Perspective/… – Simon S May 27 '15 at 19:12
• If you have sufficiently high aptitude in "real" math ability, i.e. solving problems that you have no idea how to solve when you first look at them but then you realize some trick or proof technique that will work, then one of the easiest ways to effectively prepare is to simply go through old problems and then read the solution(s) if you can't solve them, and remember all the tricks you see and/or come up with. There's only so many high school accessible math tricks that can be used to come up with problems for these contests. – user2566092 May 27 '15 at 19:36
• @user2566092 I tried doing this - trying BMO questions but soon realised I had no chance of solving it - I feel like if I just try to do the questions without any background knowledge it will be an unnecessarily arduous task. – Marek Kurczynski May 27 '15 at 19:45
• Sometimes the basic idea of the trick to use isn't obvious, because there are different variations on tricks. Still, attempting problems is in my opinion one of the best ways to prepare and get better. The preliminary US high school AIME exam problems are also good (and easier) for practicing novel problem solving and learning some tricks if you feel like you aren't ready to tackle national Olympiad or IMO problems yet. In any event, regardless of how you get to the point where you feel like you can try IMO problems, you'll probably benefit a lot at that point if you practice on old exams. – user2566092 May 27 '15 at 19:54
• I do not remember that math olympiads would require any special theory. It looked more like sport: just practice on problems of similar level. – Alexey May 27 '15 at 20:17

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.

• At the moment I just started going through Primer for Mathematical Competitions as it seems very easy and straight forward in its style. I will try to look into the recommended reading you've supplied. In regards to uni maths, what approach would be best - focusing on one topic area such as calculus and trying to go as deep as I can in that area or just trying to get the basics of undergrad maths down for the moment? – Marek Kurczynski Jun 3 '15 at 20:29
• If you do start university-level math, I would recommend Spivak's book, along with its solution manual (to be used sparingly). It's an excellent introduction to rigorous math - both in terms of theory and problem-solving. If you find that it's too difficult, then keep reading more elementary stuff. I think books 1 (Niven) and 3 (Bellman) in the Lax series are good preparation, in that they're both interesting and (both) teach you to deal with inequalities, which is something essential in rigorous calculus. For contests, you'll still need to learn geometry and number theory separately... – Keith Jun 3 '15 at 20:46
• (cont'd) and combinatorics. If it were just for math's sake and there were no contests in the picture, I would suggest focusing primarily on Spivak's book or preparation for it. – Keith Jun 3 '15 at 20:47
• This is the thing, I want to get reasonably good at the competition math but I am also weary of it getting in the way of me studying math for its own sake. I guess I will try to juggle the two in the next 2 years before uni. :) – Marek Kurczynski Jun 3 '15 at 20:50
• One of my points was not to underestimate the gain in problem-solving ability that comes from studying more advanced theory. The only problem is if you find it's too difficult, in which case you can study more elementary things for a while. – Keith Jun 3 '15 at 20:53

My personal favourite for the mathematical olympiad is "the Mathematical Olympiad Handbook; An Introduction to Problem Solving" by A. Gardiner