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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 was pretty good at math contests, but I was not good at giving advice. 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." :) –  Thomas Andrews Nov 7 '13 at 14:16
@ThomasAndrews This is a great advice for math contests –  Amr Nov 7 '13 at 14:17
This question highly depends on your math education. IMO problems are by definition international, but the way we teach math isn't. So for some people, the problems they do in class are close to IMO problems (in a sense that they know the notion that you may/should use to solve an IMO problem) and consequently in this case they only need to practice. But other people need to learn the notion before even considering doing IMO problems. –  user37238 Nov 7 '13 at 14:23
When you say, "I want to do good on IMO," does that mean you are already on your national team? Or does it mean that you want to qualify for your national team? Or do you just mean you want to be be good at solving those sorts of problems? Or are you calling a test which qualifies you to join your national team the IMO? –  Thomas Andrews Nov 7 '13 at 16:46
@Panarit, if you do not put an at sign before the name, in your case Thomas Andrews, the user is not notified of your comment. If you are looking, you have about three minutes to edit your comment to add an @ sign. Later, you can make a separate comment with the @ sign and delete this one, so there is no real rush –  Will Jagy Nov 7 '13 at 20:41
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4 Answers

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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My favorite site! –  Ahaan Rungta Nov 7 '13 at 15:38
@amWhy: Nice recommendations +1 –  Amzoti Nov 8 '13 at 4:38
Nice suggestion! –  Sami Ben Romdhane Apr 14 at 14:21
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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.

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+1 I don't understand why this answer was down-voted –  Amr Nov 7 '13 at 15:14
It is a very bad advice for math competitions. This only applies to some competitions like Putnam, where problems are, generally, designed to be easy after a one key idea is applied. In other competitions like IMO, Chinese, Vietnamese, national contests, this is not the case. In more local contests, you can even get problems that are not carefully designed with long, tedious parts in the solution. It is a bad advice to regard as wrong a path that seems to involve some volume of work. –  ABC Nov 7 '13 at 15:21
If a short advice is to be given, it should be hit the problem in every way you have. At the end, the important thing is to defeat the problem. And for training, solve solve problems, and read solutions of others. –  ABC Nov 7 '13 at 15:23
It certainly depends on the context, obviously. I'm talking US tests, where I placed top ten nationally twice (once in high school, once on the Putnam.) And I confess, when given geometry problems, I often switched to messy Cartesian coordinates to solve rather than using my limited geometric reasoning. –  Thomas Andrews Nov 7 '13 at 15:30
Thomas Andrews has good advice for math competitions. Trust me, I know. –  André Nicolas Nov 15 '13 at 9:16
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You may like to look at 'Challenge and Thrill of Pre-College Mathematics' by V Krishnamurthy, C R Pranesachar (New Age International Publishers)

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