I'm a high school student who really likes maths and I'm quite good at school. I want to start training maths by myself but I think I need some guidelines. I want to do well on IMO but I don't know what books should I read, or in what fields should I concentrate. Can you give me any advice and propose some good books about this?
Thank you !
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I suggest you take a look at the website The Art of Problem Solving.
There are links to resources, to articles, to competition preparation books, an online AoPS competition problems ("For the Win"), and more: all geared to bright students who love math, are looking for challenging problems, and it is particularly aimed to those students who participate in (or would like to start participating in) competition math.
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From my comment above:
My main advice is that there is almost always an "easy" answer. If you are heading down a path that involves a page of calculations, you are usually heading in the wrong direction. (By "easy," I don't mean "easy to find," only "easy once you've found it.")
This is, according to a comment below, probably more true for US math contests than international contests and contests in other countries.
I'd also add that you need a deep understanding of the mathematics that you know. If mathematics is just a set of facts and techniques to you, then you probably are not going to do well on math contests. It is only by understanding the facts and techniques - why they are true, why they work - that you will find creative ways to use them.
In my very first high school math contest, there was a problem to calculate the limit of the series:
$$\frac{1}{3} + \frac{2}{3^2} + \frac{1}{3^3} + \frac{2}{3^4} + \dots$$
There are loads of ways to solve this problem, if you know geometric series. I did not.
I did, however, know how to write numbers in other bases, and I did know how to turn infinite repeating decimal expansions into fractions. So I was able to solve this problem using techniques I had learned in grade school, treating the above as $$(0.121212\dots)_3$$ Even then, there were pitfalls - multiplying by $100$ and subtracting in base $10$ becomes multiplying by $(100)_3$ and subtracting, for example. If I didn't understand the above techniques, I could not have combined them correctly.
One caveat to the first part of my answer: Know your strengths. Often, the hardest problems for me on contests were geometry problems. When stuck, I'd convert the problem to an algebraic one with coordinate geometry, even though the work was messier.
The Art and craft of problem solving is very good.
http://www.amazon.com/The-Art-Craft-Problem-Solving/dp/0471135712