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Math olympiad beginner's page

This page is meant for people who don’t have much past olympiad/proof experience and are looking to get started. If you aren’t interested in proof-based problems yet, then this page is not for you. Try checking FAQ C-0 if you are totally new to math contests.

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, anyway1). Stay around for long enough, and you will get to see a lot of really amazing problems.

You may also read math contest FAQs for some more philosophical (and less concrete) advice on studying for math contests in general.

0. Syllabus#

I wrote an unofficial syllabus for math olympiads (also linked on handouts) giving some guidance on what topics appear on math olympiads.

1. First reading: the welcome letter#

For the USA Math Olympiad in 2020, the board of the USAMO prepared an invitation letter2 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:

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 the USA(J)MO unexpectedly, it’s because their proof is actually incomplete, 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:

You don’t need to get too caught up in these links; proof-writing will become more natural over time as you solve more problems. So I would 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 to practice writing in isolation.

If 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. Almost everyone gets by instead with something like the following algorithm:

  1. Write up your solution neatly.
  2. Look up the problem on AoPS contest index4; and compare your solution to those by reputable users.
  3. Edit your solution and post it on the thread. By Cunningham’s Law, wrong solutions are often exposed quite rapidly.

Together these three steps should catch “most” substantial errors. See Section B.1 of my English handout for more details about this procedure.

If you want a book to follow, the one I grew up with was Joseph J Rotman’s Journey into Mathematics: An Introduction to Proofs.3

If you like excessive information, you might also read my handout Intro to Proofs for the Morbidly Curious.

3. (For USA) 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 the 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 it is more difficult since 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.

Some possible suggestions for introductory books:

  • Geometry: My book E.G.M.O.; and more alternatives are linked at the bottom of that page.
  • The OTIS Excerpts has beginner introductions for several topics:

    • Inequalities (chapters 1-2)
    • Functional equations (chapters 3-4)
    • Combinatorics (chapters 6-9)

More possibilities (including intermediate-advanced texts not listed here) are on the links page. 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.5

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 (alphabetical):

The bottom of the recommendations page has some more suggestions for problems if this list isn’t sufficient.

Good luck and happy solving!

  1. Insomuch as contest problems have intrinsic artistic value, proof-based exams are a more versatile medium than the short-answer exams, much like how videos are more versatile than static photos. In this analogy, videos don’t make stills worthless or obsolete; they aren’t automatically better, either. But as a medium, they expand the space of ideas an artist could express, at the cost of being proportionately more work to create. 

  2. 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. 

  3. It shares my philosophy that teaching proof-based classes by force-feeding set theory notation is not particularly helpful, and instead develops proof-writing by discussing real mathematical content from geometry, number theory, etc. rather than being overly focused on bookkeeping and formalism. 

  4. I do NOT recommend using the AoPS Wiki in place of the Contest Index. The solution quality in the wiki is generally much poorer than the forum. 

  5. I used to carry a binder with printouts of the IMO shortlist and check them off as I solved them. 

Updated Sat 20 Jul 2024, 21:35:29 UTC by 7553e9c81012