This page is meant for people who don't have much past olympiad experience (or even proof experience) and are looking for a place to start.
Before all else, welcome to the olympiad scene! It's going to be hard as heck, but in my private opinion this is where all the coolest stuff is (as far as math contests go, anyways). Stay around for long enough and you will get to see a lot of really amazing problems.
You may also read math contest FAQ's for some more philosophical (and less concrete) advice on studying for math contests in general.
1. First reading: the welcome letter
For the USA Math Olympiad in 2020, the board on the USAMO prepared an invitation letter for all the qualifiers, congratulating them on their achievement and giving them some suggestions on where to begin. This letter contains:
- A few pretty carefully chosen problems (not necessarily easy!), to give people a sense of what to expect on the contest
- Fully written solutions to those chosen problems, so that you can see what a correct and complete solution is expected to look like.
- Some advice for actually taking the contest: the format of the exam, planning your time, common mistakes, etc.
You can download the letters here:
(These were written in early January 2020 before COVID-19 wreaked havoc on everything, so the contests still go by their typical name and don't mention anything specific to the belated administration that year.)
I suggest starting by reading through this letter, trying the example problems (you will probably not solve them all; we chose examples from the entire difficulty spectrum), and then comparing your work to the provided solutions. That will give you a taste of what you are getting in to.
2. Writing proofs
If you don't have experience with proof-based problems, the first thing I should say is that it is not as hard as you might think. Solving the problems completely is difficult, but if you really have a completely correct solution to a problem, it is actually pretty hard to not get full credit. I would say at least 90% of the time, when a student loses points on USA(J)MO unexpectedly, it's because their proof is actually wrong, not (just) badly-written.
Of course, you should still try to write your solutions as clearly as possible. To that end, here are some links to advice:
- How to write a solution, from Art of Problem Solving.
- Remarks on English, written by me.
- How to write proofs, by Larry W. Cusick.
You don't need to get too caught up in these links; proof-writing will become more natural over time anyways as you solve more problems. So I would actually encourage you to continue doing practice problems or reading books at the same time as you are getting used to writing; these go hand-in-hand and I actually suspect it's counterproductive to try and practice writing in isolation.
If it is possible, the best way is to have a friend or coach who can check your work and provide suggestions. But the supply of people willing to do this is admittedly very low, so most people are not so lucky to have access to feedback. The second-best thing to do is write up your solution and post it on AoPS contest index; by Cunningham's Law, wrong solutions are often exposed quite rapidly. You should also compare your solutions to existing ones.
Of course, you should set your sights on becoming self-sufficient eventually. During an exam, you do not have access to feedback or other's solutions, so you need to be able to know when you've solved a problem!
3. United States Mathematical Talent Search and USEMO
If you are in the United States, there is a nice proof-based contest called the USAMTS which is a great way for beginners to get started. Things that make the USAMTS special:
- It is a free, individual, online contest open to any students in USA.
- The problems are chosen to be quite beginner-friendly, though with a spectrum of difficulty each round.
- This contest gives you a full month to work on the problems rather than having a short time limit.
- You get some feedback on your proofs as well, not just a score.
I also run a contest called the USEMO in the fall which is also free and offers feedback, but is intended to mimic the USAMO and IMO in format and difficulty. One could try using it as "practice USAMO" in the fall.
4. Books to read
There is some more material you have to learn as well, since there are some new classes of problems (such as olympiad geometry, functional equations, orders mod p, etc.) that you will likely not have seen before from just working on short-answer contests.
To toot my own horn further, as if I haven't enough already, I wrote the following two books to be possible introductions:
- The OTIS Excerpts for everything other than geometry, and
- The textbook E.G.M.O. for classical geometry.
... but there are some more possibilities listed on the links page. Another nice general introduction is Art and Craft of Problem Solving by Paul Zeitz. You might also check Geoff Smith's advice and links.
(I really want to stress these are mere suggestions. Just because you have done X does not mean you will achieve your goals, and conversely, there are surely many fantastic resources that I have not even heard of. If you are looking for a list of materials which are "guaranteed to be enough" for solving IMO #1 and #4, then unfortunately I can't help you.)
5. Problem sources
At some point (sooner rather than later), you also need to start just working through some past problems from recent years of contests. You can think of this as encountering problems in the wild. (I used to carry a binder with printouts of the IMO shortlist and check them off as I solved them.)
In case you didn't know already, on Art of Problem Solving there is an extensive archive of past problems from basically every competition under the sun, together with community-contributed solutions. The supply of problems here is inexhaustible.
Here are some particular contests I like (roughly ascending order of difficulty):
- Canada national olympiad
- European Girls Math Olympiad
- IMO and IMO Shortlist
- USA Team Selection Tests
The bottom of the recommendations page has some more suggestions for problems if this list isn't sufficient.
Good luck and happy solving!