These are FAQ's about math contests and particularly how to go about training for them.
1. How should I prepare for math contests?
One-sentence answer: do lots of problems just above your current ability, and spend some of time on reflection. Full answer: this blog post.
It's worth pointing out that no one has a silver bullet: there's no known study method $X$ (let alone a specific handout/book) such that students following $X$ consistently do well on USAMO. (Reason: the existence of $X$ is not a Nash equilibrium.) There are plenty of strategies which obviously don't work like "do nothing" and "read solutions without trying any problems", but beyond that anything reasonable is probably fine.
If you had to force me to say what I thought was the biggest predictor of success, I would say it is whether you think about math in the shower. That means you both enjoy your work and are working on things in the right difficulty range.
2. Which books/handouts/materials should I use?
As I said it probably doesn't matter too much which ones you choose, as long as you exercise some basic common sense.
With that disclaimer, here are some possible suggestions for math olympiads.
Younger students (preparing for AMC/AIME) would likely benefit from books or classes from Art of Problem Solving, like Volume 2.
3. How do I get better at Euclidean geometry?
4. Is it possible for me to go from $X$ level to $Y$ level in $Z$ time?
I probably don't know much more than you do.
Predicting improvement in math contests is a lot like trying to predict the stock market. We have some common sense, but no one really knows much more than that.
5. How do I learn to write proofs?
I don't think there's actually a leap between computation and proof-writing, and I actually suspect that thinking proof-writing is hard is most of what makes it hard. Reasons it might appear hard:
- US students get basically zero exposure to proofs (even in the contest world), and are led to think that it's something mythical that's above them, when in fact it's merely just being forced to explain your solution on paper.
- Having zero experience also means that you might not present your ideas clearly or violate some unspoken rules on style.
- Some classes of problems (like inequalities) don't appear at all until the olympiad level, so students have to learn how to write a proof (fairly easy) while simultaneously learning a new class of problems (hard).
- Read the proofs to problems you think you've solved (on AoPS, or official solutions). Note that these don't have to be from proof contests! The official solutions to any decent contest would all pass as proofs.
- Try writing up proofs to problems you think you've solved, and
- Get feedback on these solutions (from a mentor, on the forums, etc.).
The USA Mathematical Talent Search is also another good option. It is a free mathematics competition open to all United States middle and high school students, which gives you about one month to produce full solutions to a set of five problems.
6. Am I ready to do $X$ level problems, read book $Y$, etc.?
The correct thing to do is just try it out (e.g. try some problems from a past $X$ paper, read a chapter from $Y$, etc.) and see how it feels. You all know what it feels like when something is too easy (think middle school math class): you feel like you're doing the problems for the sake of doing the problems rather than actually learning. You all know what it feels like when something is too hard: the dreaded "I have no clue what's going on".
Anything not too close to either extreme is probably fine. If in doubt, I recommend picking whatever is most enjoyable.
7. Should I read $X$ book versus $Y$ book, spend $M$ hours versus $N$ hours per week, etc.?
My gut feeling is that the effect size of this is sufficiently small that (i) no one knows a definitive answer, (ii) the answer is likely to depend on the person, and (iii) it's rounding error compared to the actual final result. Therefore I think the correct thing to do is try both, see which one you like better, and just go with that, without worrying about whether it is "right".
8. How do I make fewer careless errors?
There's a nice article on AoPS that addresses most of what I have to say. Here are just a few additional remarks.
We're not kidding when we say to be neat: it really is hard to think if your scratch paper is a bunch of clutter. To get an idea of what my scratch work looks like, here is my scratch paper from the 2013 AIME. Some things worth noting from it are:
- Every problem is labelled on its own page (or multiple pages).
- Diagrams are very large, often taking up half the page.
- Mistakes are simply "struck out" rather than scribbled out or halfheartedly erased.
- The final answer on each page is boxed for easy reference later.
Also, don't misread questions, don't rush, etc.
Anyways, I admit there's not always a whole lot you can do about it. 2013 was the first year where I was able to look at a problem and basically know how to do it within one or two minutes; this left me a lot of time for computation, and consequently I made very few errors as compared to 2011 or 2012. In other words, as you get better at problem-solving you'll naturally become less likely to make careless errors as well. (At least that's how it turned out for me.)
See also: Against Perfect Scores.
9. Should I take the time to write up full solutions to problems I solve in training?
It depends on who you ask; you'll get different advice. On the one hand, writing the solution completely is actually really helpful (cleaning up solution, noticing errors, just getting to know things better). On the other hand, can also be very time-consuming (time that could be spent solving a different problem). So you get some varying answers and it may just depend on the person.
That said, I think it is usually a mistake to write nothing at all. I require my students to at least write down a couple sentences on the main idea of their solution --- so that, if they read the solution a few years later, they would understand the main idea, and basically know how to work out the rest of the details from there. This process of having to summarize the key ideas is also helpful, and takes so little time that there's no excuse not to do it (in my opinion).
Of course, a lot of strong students still do full write-ups anyways, and I do so myself, just out of a habit of trying to do things well.
10. Can you solve $X$ problem for me?
Probably not. If you send me a problem, usually I will at least read it. If I have seen it before or can quickly see how to do it, I will generally be nice enough to write back and outline or link the solution. But otherwise I will likely be too embarrassed to admit I don't have time to work on every problem that students send me, and simply archive your message.