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These are FAQs about math contests and particularly how to go about training for them.

C-0. I have zero experience with math contests. Where should I start?#

For the first several weeks, the most important thing is exposure. There is a time and place to think about study strategy, test-taking strategy, and so on. However, it is not right now. Instead, I recommend the famous Go proverb: “lose your first 50 games as quickly as possible”.

When you first start, you should do some subset of the following things for several weeks until you have a picture of what the landscape looks like:

1. Pick up any standard textbook like AoPS v1 or AoPS v2 to work through, so you learn some of the standard theory that is tested in math contests. Or, search around the web; there are lots of other sources.
2. Go through some past problems from previous contests. For example, if you are in the United States, then you can find all past AMC problems from the Contests section under “USA Contests”.¹ The same should be true for many other countries. Pick any year (later years are harder) and work through some problems without a time limit. For now, just pick whichever problems look most interesting. When you hit one you can’t solve for an hour, read the solution. Every contest will have problems ranging from super easy to really difficult, so find the right ones.
3. Rope some friends into learning with you. It’s more fun that way, and you can learn from each other. It’s okay to have different levels of experience.

You should repeat these steps until you have some comfort with the kinds of problems that appear.

Here are a few things that will be surprising to people who previously have mostly done school-based mathematics:

• Some contest topics, like combinatorics and number theory, are nearly nonexistent in most schools. Consequently, it is usually not possible to score well using just school math (unlike SAT). You must learn additional totally new material.

• Even when you know relevant theorems, competition problems can still demand some insight and intuition in order to solve them. This means surface-level understanding is greatly penalized (e.g. memorization). Therefore, you need to learn material with a greater depth than school. (For example, in many American schools, you are usually given a “recipe” for each exercise to carry out; in competitions, you often invent recipes on the fly.)

• As you get experience, you will automatically start to know what deep understanding feels like. (At the moment this is probably just zen monk talk.) For now, you should just be aware that you will see many, many problems which you can’t solve, where you read the solution and ask, “how was I supposed to think of that?”. This is okay and expected: it’s not because you’re dumb, it’s because you are learning. (And this never goes away!) Stay determined.

• Math competitions are notoriously difficult. In any exam, getting even 1/3 or 1/2 of the problems correct is already an accomplishment. You will be competing against some students who have been seriously preparing for many years. Even then, many of these students will not do well; like in real life, hard work does not automatically guarantee success.

  If this sounds discouraging, it isn’t meant to be. What it means that you have to enjoy the process of learning itself, not merely a means to an end. That’s why contests exist anyway.

Good luck!

C-1. How should I prepare for math contests?#

One-sentence answer: do lots of problems just above your current ability, and spend some 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.

This is all generic advice, so some of you will email me asking for more targeted suggestions. If you do, please read the special instructions for study advice.

C-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.

C-3. How do I get better at Euclidean geometry?#

For general advice, see my advice, Geoff Smith’s advice, etc.

For specific materials, see my own handouts or Yufei Zhao’s, among others. And of course, my geometry textbook.

C-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.

C-5. How do I learn to write proofs?#

There’s an extremely verbose answer in the handout Intro to Proofs for the Morbidly Curious but here’s a shorter answer:

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).

To get you started, try reading this article, or perhaps this article. After this, my advice is

1. 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.
2. Try writing up proofs to problems you think you’ve solved, and
3. Get feedback on these solutions (from a mentor, on the forums, etc.).

If you want to follow a book, the one I grew up with was Joseph J Rotman’s textbook. 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. Courses on graph theory are often good introductions to proofs as well.

If you are a United States middle or high school student, The USA Mathematical Talent Search is also nice option. It is a free mathematics competition which gives you about one month to produce full solutions to a set of five problems.

C-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.

C-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”.

C-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.

Anyway, I admit there’s not always a lot you can do about it. 2013 was the first year when 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.

C-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. The obvious downside is that it’s time-consuming (time that could be spent solving a different problem). But I think the upsides are understated; see FAQ C-24. For this reason, a lot of strong students still do full write-ups anyway, especially those who have a “do things well” habit.

I’d say use your judgment. If your solution involves a long but straightforward arithmetic calculation, for example, then transcribing that full calculation into a text editor is likely unnecessary. Conversely, if you have a lot of moving parts or confusing logic, writing those out can help you make sure it checks out.

That said, I think it is usually a mistake to write nothing at all. For OTIS, I require my students to at least write down a couple sentences on the main idea of their solution — so that, if they read this 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).

C-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.

C-11. I can’t be bothered to learn directed angles. Can I just use normal angles?#

As I allude to in the directed-angles handout you could probably get away with it, because it is common practice to not deduct for configuration issues (although it does happen from time to time). So, if you really hate it, you can ignore it.

However, in the words of Ankan Bhattacharya:

You should learn directed angles, not necessarily because they deal with inessential configuration issues, but because once you know them angle chasing with directed angles becomes easier than normal angle chasing.

I have the same opinion. I initially used directed angles anyway because I cared about having complete correct solutions even if I didn’t lose points for it. But then I realized it’s actually easier once you get used to it, because you don’t need to keep track of $x$ vs $180^{\circ}-x$ anymore. So I don’t think I would go back to using normal angles even if I could.

C-12. How long should I spend on an olympiad problem before I give up?#

You should have a lower bound (1 hour is a good rule of thumb). But I think for the upper bound, it’s often better to go by a feeling of how much progress you’ve made rather than a hard time.

What I mean by this is, you can spend many hours on a problem and still learn a lot from it if you’re making progress or finding new things the whole time. In other words, if you still have steam, you should keep going. But conversely, if you stare at a problem for three hours with no clue what to do the entire time, it’s probably okay to give up. There are more problems to do.

So for the upper bound, I think you can trust your per-problem judgment a bit more. It’s not a bad rule of thumb, honestly, to give up once the problem stops being fun (and you are over the lower bound). That’s usually a good indicator for me when I’m not getting anywhere, at least. (And at the risk of stating the obvious, I recommend reading the solution to a problem if you decide to stop working on it.)

It’s also encouraged to have a few problems you’re thinking about at once rather than literally one at a time, so that you can switch between problems as you see fit (e.g. if you are feeling stuck on one and want a break).

C-13. Do I need to know X topic for math olympiads?#

Read the syllabus linked on the beginner page.

C-14. In geometry, can I only use synthetic methods and not use coordinates? Or vice versa? Which coordinate system is most useful?#

Wrong question. You should always use the tool most suited to that problem. Some problems are easier to solve with coordinates than synthetic methods. Some problems are easier to solve with synthetic methods than coordinates. Some are totally intractable with one approach, and so on. Therefore, you should aim to be versatile in choosing approaches. In particular, knowing multiple methods well pays off.

Lines of attack also should be used together, rather than individually. The following story is common for me: I am working on some geometry problem, whether for fun or vetting it for a contest. I quickly make some synthetic reduction to reach a problem that I could complex bash if I wanted to, but which is ugly. Of course, I am lazy and don’t want to carry out the ugly computation, so I keep trying to make more reductions to get a better setup for complex numbers. Then after some more reductions, I found that I have discovered the synthetic solution anyway! It seems moving between perspectives simultaneously helps you solve more problems than you could trying to only use a subset.

C-15. On a 3-problem, 4.5-hour exam, should I try all three problems?#

Yes. Here is why:

1. In math contest world, it is extremely common for problem difficulty to be rather volatile. Sometimes the test difficulty is out-of-order (see IMO 2021). Sometimes problem 2 is ridiculously hard (see IMO 2017). Sometimes the test has three problems that are more like medium (see USAMO 2022). Sometimes problem 6 is secretly a medium problem (see IMO 2016). The list of weird scenarios never ends. (The MOHS chart has concrete but imperfect data.)
2. Moreover, problem difficulty is personal and differs between people. (In cliché format, you have your own unique individual strengths and weaknesses.) So, every contest you will see score distributions like 070 or even 007 (“James Bond”).
3. Even in cases when problems 3 and 6 are genuinely difficult, partial marks on them may often be accessible. See IMO 2020 for an example.
4. Personally, I suspect the attitude that a late-placed problem is definitely out of reach and there is no point reading them is counterproductive in the long term.

I believe this so firmly that I actually often give the following suggestion:

Try all 3 problems in reverse order, for 20 minutes each, at the start of the test.

I’ve said this in various forms not just in OTIS, but also when I teach at MOP or other places, etc. Unsurprisingly, people find this really hard advice to swallow (see C-20 for a bit of related discussion).

I should defeat the straw man right away: this is vastly different from “do the problems in reverse order”. It is, “start the test by trying problem 3 for 20 minutes, then problem 2 for 20 minutes, and then do what you want”.

The reason I give this advice is no matter how much I beg and plead students to at least look at all three problems seriously, I realize that it’s psychologically difficult to do so during a high-pressure environment, especially if you happen to be tilted because problem 1 was sent in by Holden. So at some point I gave up on asking nicely, and now I just give this more blunt actionable piece of advice to sort of force people to do it.

It’s not for everyone. You should try it on a few practice exams and see if it works for you.

C-16. What grade should I aim to reach X goal by?#

You should just do your best each year.

I see so many people saying things like “I would try to make JMO by 8th grade to make MOP in 10th” or “I didn’t make JMO in 9th grade, so now I’m never going to make MOP” and other comments like this. Here’s some advice you didn’t ask for: you’ll make it farther if you lose that mindset.

Why? The problem with that kind of thinking is you’re treating time as your enemy. You put yourself in a mindset where every year is a race against the clock, and if you fall behind your quota (that someone on the Internet put for you), then all hope is lost. I think this sort of outlook is so negative and results-oriented that you will actually perform worse because of it. You can become so upset about the results of 9th grade that you can’t study in 10th grade. Instead, I think you should try to treat the time as your friend. If you don’t do well in 2020, that’s okay, you have lots of time to try again in 2021 and 2022 and later on. You don’t have to try and extrapolate years into the future — because the leading term for your 2021 score is not your 2020 score, it’s what you do between now and then.

You will find it’s impossible to learn well if you hang a clock over your head.

C-17. Do I need to make a study plan?#

You can if you want to, but in my opinion it’s not that important. I certainly never made a plan as a student.

The reason I have somewhat dim views of study plans is that energy management is more important than time management. In general, it is a lot better to be spending a lot of time exploring whatever is catching interest, than to try and lay out some fixed plan and then not really follow through with it. So even though in theory planning seems like it would be helpful, in practice the energy spent crafting then adhering to a plan could easily outweigh benefits. (One can always start with a lot of energy, and then the energy quickly dies out.)

I think the only concern might be if a student is afraid of a particular area and tries to ignore it completely (geometryyyyyyy). As long as you avoid this trap, I hereby give you license to work on whatever you feel like, planned or not.

C-18. Do I have to do EVERY problem from X?#

No.

C-19. Should I practice with hand-drawn or digital geo diagrams?#

I’ve heard different advice from this from different people across the years, ranging from never to always. Proponents of hand-drawn diagrams say that it’s important to not become reliant on easily generated computer figures because they are not available during competitions. Proponents of Geogebra say that practicing with perfect diagrams is good for building a sense of where things should be, plus it saves you time, so you can do more problems. I don’t get the feeling there is a consensus as to what the common wisdom is.

My suggestion is to just do a mix of both with the exact balance up to personal taste and convenience. I don’t think it matters too much as long as you don’t neglect one completely. But in the same vein as C-17, if you (say) tell yourself you’re going to only use paper diagrams because X said so, and then you end up practicing less because it’s inconvenient to have a pencil and compass out during your deadly-boring AP Java class, that’s your loss.

C-20. How do I handle stress or pressure?#

I don’t think it ever goes away completely. Shows that you care. If you read my diary of IMO 2014, there was a point during day 1 where I became so nervous that my hands wouldn’t stop shaking, and I couldn’t carry out a certain trig calculation correctly for an hour or so. (I dealt with this by standing up for a few minutes; when I sat back down my hands had stopped shaking, and I did the entire calculation from scratch in 10 minutes.)

Nevertheless, here is one possible piece of advice. There’s a saying that “if it’s hard for you, it’s hard for everyone”. People often interpret this as a form of encouragement, trying to say that you’re better than you think you are. I do not like this interpretation, because I think it makes the saying into a statement about egos, when it should be a statement about exam design.

You have a skill level that you’ve been working on all year. In addition, you will have some individual variance based on the actual day. But your given-day individual variance is completely dwarfed by a larger term: the difficulty of the problem, which often varies wildly from year to year. In many cases, different years of the same exam are not even close to comparable. See C-15 for concrete discussion, or the MOHS chart if you like colors.

Thus, if you find yourself surprised at being unable to solve a problem, your reaction should be something like “wow, this year’s P2 is harder than usual”.

Do not misinterpret the difficulty of an exam as a skill issue. This is one of the most common ways to underperform. I have seen so many students do poorly on an exam because they thought a certain problem must be easy or hard based on its problem number, only finding out later they were wrong, and then blaming their low score on the test designer (a really lame excuse).

C-21. How should I prepare for the Putnam?#

For the most part, preparation for Putnam looks similar to that for USAMO, except that Euclidean geometry is absent, while abstract and linear algebra, calculus, and a bit of real analysis are fair game. Most non-geometry resources on this website should work equally well for Putnam. For the undergraduate topics, reading a bit of Napkin may be helpful for grounding if you haven’t seen those areas before.

Format-wise, the most stark difference, in my opinion, is that the time controls for the Putnam are much faster. In each session, you have six problems to solve in three hours. (In contrast, the IMO gives three problems to solve in 4.5 hours.) Moreover, the difficulty ramp-up from 1 to 6 is less steep than you might be used to if you’ve only done high-school olympiads before. So, the skill of deciding in five minutes whether to continue working on a problem or not is significantly more important.

Finally, disclaimer: these days, many top-scoring Putnam students do not seriously prepare for the Putnam during the school year (and thus are relying on high-school training). Especially if you’re someone who already did math olympiads in high school, I encourage you to not place more weight on the Putnam than it deserves.

C-22. For X contest, about how much do I usually need to get Y medal/award?#

If X is a USA contest, you can often extrapolate some values from the statistics posted on my problems archive. And of course for the IMO, you have full data. So you should be able to work out an estimate for yourself. For other contests, maybe you’ll have to ask around if the contest organizer doesn’t have statistics easily available.

However, I want to warn you: I do not recommend taking too seriously the arithmetic mean or median of whatever statistic you are looking at. As I mentioned in C-20, the fluctuation in whatever random variable you are measuring is large enough that averages are often meaningless. You may hurt yourself in your own confusion if you approach a contest with a fixed mindset of, say, “I need to get three problems for silver”, and then the test (really, individual problems on the test) ends up being easier or harder than the worthless expected value that you worked out beforehand.

C-23. Since contests get harder over time, should I only work on recent years?#

At time of writing (2023), I’d say anything post-2000 is probably fine. It’s probably true that 2000-2010 is on average easier than 2010+, but not by enough that I would really care, especially if you’re a beginner. (In fact, for beginners, the 2000-2010 era of problems could actually be better fits than the 2010+ era on average, because on average they assume less training and background knowledge.)

C-24. Why are you such a stickler about write-ups? You get 7 points at a contest as long as the solution is correct, right?#

I think that if you try to divorce the solving process from the writing process, it will damage your ability to do both. I treat the writing process as a chance to improve and clean the solution rather than merely a chore of transcribing it.

Many of you have this idea that you should solve the problem first and then just transcribe what’s in your head onto paper. You indeed can get away with this at IMO because there’s no deduction for garbage-but-correct papers. But your solutions will be both less comprehensible and mathematically less elegant if you work this way, even if they are technically correct. (And your error rate will be higher as well.)

Whereas if you are trying to write the solution well (not just for the 7 points like many of you have been trained to do), you should often find that the content of your solution will be different from what was in your head before you started writing. If you watch me do write-ups live on stream you’ll see this. It’s only during the so-called “write-up” phase that a lot of the claims are restructured or reordered, or steps are consolidated and simplified. (Or worse still, I realize the solution I had in my head doesn’t actually work. Sometimes it’s a minor patch, but sometimes the whole thing is broken.)

Essay-writing and painting work similarly. Paul Graham wrote the following quote about essays

If all you want to do is figure things out, why do you need to write anything, though? Why not just sit and think? Well, there precisely is Montaigne’s great discovery. Expressing ideas helps to form them. Indeed, helps is far too weak a word. Most of what ends up in my essays I only thought of when I sat down to write them. That’s why I write them.

and the following about painting:

Another example we can take from painting is the way that paintings are created by gradual refinement. Paintings usually begin with a sketch. Gradually the details get filled in. But it is not merely a process of filling in. Sometimes the original plans turn out to be mistaken. Countless paintings, when you look at them in xrays, turn out to have limbs that have been moved or facial features that have been readjusted.

It does kind of suck that on actual high-school exams, you have to write solutions with pen and paper. These days I type everything, so it’s much easier to edit work-in-progress. For example cut-and-paste is a godsend. So by all means, during a contest, focus on just getting full marks. But when you’re at home practicing, I encourage you to not settle for “technically correct” because I think you’ll learn more that way.

C-25. Why don’t you have any olympiad algebra book recommendations?#

Because “olympiad algebra” is a hodgepodge of unrelated topics. (And confusing things further², it is largely unrelated to abstract/modern algebra as well.)

Examples of things that get thrown into the massive umbrella “algebra” include:

• A bunch of functional equations that deal with manipulating artificial expressions like $f(x+f(x+y))+f(xy)$ with equals signs.
• Some rarer (but more interesting to me personally) functional equations that say something conceptually like USA TST 2021/3 rather than just being ad-hoc manipulation.
• Really exact inequalities that come down to manipulating an artificial expression like $\sum \frac{ab+1}{(a+b)^2}$ using AM-GM/Schur/Holder and friends.
• Problems involving comparing the size of two quantities and using big-O asymptotic reasoning or similar, like Shortlist 2002 A2 and IMO 2016/5, rather than a really precise AM-GM or similar.
• Other problems that touch on ideas in real analysis like USA TST 2020/1, USEMO 2023/2, or USA TSTST 2021/2.
• Problems about polynomials with integer, real, or complex coefficients, which can be either analytic in nature or more algebraic (e.g. Vieta), or both.
• Miscellaneous problems about manipulating more random artificial expressions even though they’re not an FE or inequality (in OTIS, we call this the “Alg Manip” unit).
• Combinatorics problems that got tossed into the algebra list anyway.
• Number theory problems that got tossed into the algebra list anyway.
• Algebra problems that got tossed into the number theory list because they happened to use the word “integer”.

So if you are one of the countless students who has written to me the question “how do I get better at olympiad algebra”, the first step to enlightenment is realizing there’s no such thing. I’d say it’s better to instead focus on a particular sub-skill or idea with real cohesion.

#AlgebraIsntReal, fight me.

C-26. My big important MO is next week, what should I do?#

Wind down.

In general, usually it’s more important to be rested (e.g. sleeping enough) and have a clear head in the last few weeks leading up to the contest, rather than trying to do additional math practice. Math olympiads are meant to be a year-round studying.

(I know some past students who try to increase intensity in the last week, but it ends up making them more nervous, hence net harmful to their score.)