# On choosing exercises

Finally, if you attempt to read this without working through a significant number of exercises (see §0.0.1), I will come to your house and pummel you with [Gr-EGA] until you beg for mercy. It is important to not just have a vague sense of what is true, but to be able to actually get your hands dirty. As Mark Kisin has said, “You can wave your hands all you want, but it still won’t make you fly.”

— Ravi Vakil, The Rising Sea: Foundations of Algebraic Geometry

When people learn new areas in higher math, they are usually required to do some exercises. I think no one really disputes this: you have to actually do math to make any progress.

However, from the teacher’s side, I want to make the case that there is some art to picking exercises, too. In the process of writing my Napkin as well as taking way too many math classes I began to see some patterns in which exercises or problems I tended to add to the Napkin, or which exercises I found helpful when learning myself. So, I want to explicitly record some of these thoughts here.

## 1. How not to do it

So in my usual cynicism I’ll start by saying what I think people typically do, and why I don’t think it works well. As far as I can tell, the criteria used in most classes is:

1. The student is reasonably able to (at least in theory) eventually solve it.
2. A student with a solid understanding of the material should be able to do it.
3. (Optional) The result itself is worth knowing.

Both of these criteria are good. My problem is that I don’t think they are sufficient.

To explain why, let me give a concrete example of something that is definitely assigned in many measure theory classes.

Okay example (completion of a measure space). Let ${(X, \mathcal A, \mu)}$ be a measure space. Let ${\overline{\mathcal A}}$ denote all subsets of ${X}$ which are the union of a set in ${\mathcal A}$ and a null set. Show that ${\overline{\mathcal A}}$ is a sigma-algebra there is a unique extension of the measure ${\mu}$ to it.

I can see why it’s tempting to give this as an exercise. It is a very fundamental result that the student should know. The proof is not too difficult, and the student will understand it better if they do it themselves than if they passively read it. And, if a student really understands measures well, they should find the exercise quite straightforward. For this reason I think this is an okay choice.

But I think we can do better.

In many classes I’ve taken, nearly all the exercises looked like this one. I think when you do this, there are a couple blind spots that sometimes get missed:

• There’s a difference between “things you should be able to do after learning Z well” and “things you should be able to do when first learning Z“. I would argue that the above example is the former category, but not the latter one — if a student is learning about measures for the first time, my first priority would be to make sure they get a good conceptual understanding first, and in particular can understand why the statement should be true. Then we can worry about actually proving it.
• Assigning an exercise which checks if you understand X is not the same as actually teaching it. Okay exercises can verify if you understand something, great exercises will actively help you understand it.

## 2. An example that I found enlightening

In contrast, this year I was given an exercise which I thought was so instructive that I’ll post it here. It comes from algebraic geometry.

Exercise: The punctured gyrotop is the open subset ${U}$ of ${X = \mathrm{Spec} \mathbb C[x,y,z] / (xz, yz)}$ obtained by deleting the origin ${(x,y,z)}$ from ${X}$. Compute ${\mathcal O_X(U)}$.

It was after I did this exercise that I finally felt like I understood why distinguished open sets are so important when defining an affine scheme. For that matter, it finally clicked why sheaves on a base are worth caring about.

I had read lots and lots of words and pushed symbols around all day. I had even proved, on paper already, that ${\mathcal O(U \sqcup V) = \mathcal O(U) \times \mathcal O(V)}$. But I never really felt it. This exercise changed that for me, because suddenly I had an example in front of me that I could actually see.

## 3. Some suggested additional criteria

So here are a few suggested guidelines which I think can help pick exercises like that one.

### A. They should be as concrete as possible.

This is me yelling at people to use more examples, once again. But I think having students work through examples as an exercise is just as important (if not more) than reading them aloud in lecture.

One other benefit of using concrete examples is that you can avoid the risk of students solving the exercise by “symbol pushing”. I think many of us know the feeling of solving some textbook exercise by just unwinding a definition and doing a manipulation, or black-boxing some theorem and blindly applying it. In this way one ends up with correct but unenlightening proofs. The issue is that nothing written down resonates with System 1, and so the result doesn’t get internalized.

When you give a concrete exercise with a specific group/scheme/whatever, there is much less chance of something like that happening. You almost have to see the example in order to work with it. I really think internalizing theorems and definitions is better done in this concrete way, rather than the more abstract or general manipulations.

### B. They should be enjoyable.

Math majors are humans too. If a whole page of exercises looks boring, students are less likely to do them.

This is one place where I think people could really learn from the math contest community. When designing exams like IMO or USAMO, people fight over which problems they think are the prettiest. The nicest and most instructive exam problems are passed down from generation to generation like prized heirlooms. (Conveniently, the problems are even named, e.g. “IMO 2008/3”, which I privately think helps a ton; it gives the problems a name and face. The most enthusiastic students will often be able to recall where a good problem was from if shown the statement again.) Imagine if the average textbook exercises had even a tenth of that enthusiasm put into crafting them.

Incidentally, I think being concrete helps a lot with this. Part of the reason I enjoyed the punctured gyrotop so much was that I could immediately draw a picture of it, and I had a sense that I should be able to compute the answer, even though I wasn’t experienced enough yet to see what it was. So it was as if the exercise was leading me on the whole way.

For an example of how not to do it, here’s what I think my geometry book would look like if done wrong.

### C. They should not be too tricky.

People are always dumber than you think when they first learn a subject; things which should be obvious often are not. So difficulty should be used in moderation: if you assign a hard exercise, you should assume by default the student will not solve it, so there better be some reason you’re adding some extra frustration.

I should at this point also mention some advice most people won’t be able to take (because it is so time-consuming): I think it’s valuable to write full solutions for students, especially on difficult problems. When someone is learning something for the first time, that is the most important time for the students to be able to read the full details of solutions, precisely because they are not yet able to do it themselves.

In math contests, the ideal feedback cycle is something like: a student works on a problem P, makes some progress (possibly solving it), then they look at the solution and see what they were missing or where they could have cleaned up their solution or what they could have done differently, et cetera. This lets them update their intuition or toolkit before going on. If you cut out this last step by not providing solutions, you lose the only real chance you had to give feedback to the student.

## 4. Memorability

I have, on more occasions than I’m willing to admit, run into the following situation. I solve some exercise in a textbook. Sometime later, I am reading about some other result, and I need some intermediate result, which looks like it could be true but I don’t how to prove it immediately. So I look it up, and then find out it was the exercise I did (and then have to re-do the exercise again because I didn’t write up the solution).

I think you can argue that if you don’t even recognize the statement later, you didn’t learn anything from it. So I think the following is a good summarizing test: how likely is the student to actually remember it later?

# Understanding with System 1

Math must be presented for System 1 to absorb and only incidentally for System 2 to verify.

I finally have a sort-of formalizable guideline for teaching and writing math, and what it means to “understand” math. I’ve been unconsciously following this for years and only now managed to write down explicitly what it is that I’ve been doing.

(This post is written from a math-centric perspective, because that’s the domain where my concrete object-level examples from. But I suspect much of it applies to communicating hard ideas in general.)

## S1 and S2

The quote above refers to the System 1 and System 2 framework from Thinking, Fast and Slow. Roughly it divides the brain’s thoughts into two categories:

• S1 is the part of the brain characterized by fast, intuitive, automatic, instinctive, emotional responses, For example, when you read the text “2+2=?”, S1 tells you (without any effort) that this equals 4.
• S2 is the part of the brain characterized by slow, deliberative, effortful, logical responses; for example, S2 is used to count the number of words in this sentence.

(The link above gives some more examples.)

The premise of this post is that understanding math well is largely about having the concept resonate with your S1, rather than your S2. For example, let’s take groups from abstract algebra. Then I claim that

$G = \{ a/b \mid a,b \text{ odd integers} \}$

is a group under the usual multiplication. Now, if you have a student who’s learning group theory for the first time, the only way they could see this is a group is to compare it against a list of the group axioms, and have their S2 verify them one by one. But experienced people don’t do this: their S1 automatically tells them that $G$ “feels” like a group (because e.g. it’s closed and doesn’t have division-by-zero issues).

I think this S1-level understanding is what it means to “get it”. Verifying a solution to a hard olympiad problem by having S2 check each individual step is straightforward in principle, albeit time-consuming. The tricky part is to get this solution to resonate with S1. Hence my advice to never read a solution line by line.

## Writing for S1

What this means is that if you’re trying to teach someone an idea, then you should be focusing on trying to get their S1 to grasp it, rather than just their S2. For example, in math it’s not enough to just give a sequence of logical steps which implies the result: give it life.

Here are some examples of ways I (try to) do this.

First, giving good concrete examples. S1 reacts well when it “sees” a concrete object like $G$ above, and can see some intuitive properties about it right away. Abstract “symbol-pushing” is usually left to S2 instead.

Similarly, drawing pictures, so your S1 can actually see the object. On one extreme end, you can write something like “a point $S$ lies on the polar of $T$ if and only if $T$ lies on the polar of $S$”, but it’s much better to just have a picture:

You can even do this for things that aren’t really geometrical in nature. For example, my Napkin features the following picture of cardinal collapse when forcing.

Third, write like you talk, and share your feelings. S1 is emotional. S1 wants to know that compactness is a good property for a space to have, or that non-Noetherian rings are way too big and “only weirdos care about non-Noetherian rings” (just kidding!), or that ramified primes are the “finitely many edge cases” and aren’t worth worrying about. These S1 reactions you get are the things you want to pass on. In particular, avoid standard formal college-textbook-bleed-your-eyes-dry-in-boredom style. (To be fair, not all textbooks do this; this is one reason why I like Pugh’s book so much, for example.)

Even the mechanics on the page can be made to accommodate S1 in this way. S1 can’t read a wall of text; S2 has to put in effort to do that. But S1 can pick out section headers, or bolded phrases like this one, and so on and so forth. That’s why in Napkin all the examples are in separate red boxes and all the big theorems are in blue boxes, and important philosophical points are typeset in bold centered green text. This way S1 naturally puts its attention there.

## But do not force it

On the flip side, if you’re trying to learn something, there’s a common failure mode where you try to keep forcing S2 to do something unnatural (rather than trying to have S1 figure it out). This is the kind of thing when you don’t understand what the Chinese Remainder Theorem is trying to say, so you try to fix this by repeatedly reading the proof line by line, and still not really understanding what is going on. Usually this ends up in S2 getting tired and not actually reading the proof after the third or fourth iteration.

(For the Chinese remainder theorem the right thing to do is ask yourself why any arithmetic progression with common difference 7 must contain multiples of 3: credits to Dominic Yeo again for that. I’m not actually sure what you’re supposed to do when stuck on math in general. Usually I just ask my friends what is going on, or give up for now and come back later.)

Actually, I really like the advice that SSC mentions: “develop instincts, then use them”.

# MOP should do a better job of supporting its students in not-June

Up to now I always felt a little saddened when I see people drop out of the IMO or EGMO team selection. But actually, really I should be asking myself what I (as a coach) could do better to make sure the students know we value their effort, even if they ultimately don’t make the team.

Because we sure do an awful job of being supportive of the students, or, well, really doing anything at all. There’s no practice material, no encouragement, or actually no form of contact whatsoever. Just three unreasonably hard problems each month, followed by a score report about a week later, starting in December and dragging in to April.

One of a teacher’s important jobs is to encourage their students. And even though we get the best students in the USA, probably we shouldn’t skip that step entirely, especially given the level of competition we put the students through.

So, what should we do about it? Suggestions welcome.

# Undergraduate Math 011: a firsT yeaR coursE in geometrY

tl;dr I parodied my own book, download the new version here. People often complain to me about how olympiad geometry is just about knowing a bunch of configurations or theorems. But it recently occurred to me that when you actually … Continue reading

# I switched to point-based problem sets

It’s not uncommon for technical books to include an admonition from the author that readers must do the exercises and problems. I always feel a little peculiar when I read such warnings. Will something bad happen to me if I don’t do the exercises and problems? Of course not. I’ll gain some time, but at the expense of depth of understanding. Sometimes that’s worth it. Sometimes it’s not.

— Michael Nielsen, Neural Networks and Deep Learning

## 1. Synopsis

I spent the first few days of my recent winter vacation transitioning all the problem sets for my students from a “traditional” format to a “point-based” format. Here’s a before and after.

Technical specification:

• The traditional problem sets used to consist of a list of 6-9 olympiad problems of varying difficulty, for which you were expected to solve all problems over the course of two weeks.
• The new point-based problem sets consist of 10-15 olympiad problems, each weighted either 2, 3, 5, or 9 points, and an explicit target goal for that problem set. There’s a spectrum of how many of the problems you need to solve depending on the topic and the version (I have multiple difficulty versions of many sets), but as a rough estimate the goal is maybe 60%-75% of the total possible points on the problem set. Usually, on each problem set there are 2-4 problems which I think are especially nice or important, and I signal this by coloring the problem weight in red.

In this post I want to talk a little bit about what motivated this change.

## 2. The old days

I guess for historical context I’ll start by talking about why I used to have a traditional format, although I’m mildly embarrassed at now, in hindsight.

When I first started out with designing my materials, I was actually basically always short on problems. Once you really get into designing olympiad materials, good problems begin to feel like tangible goods. Most problems I put on a handout are ones I’ve done personally, because otherwise, how are you supposed to know what the problem is like? This means I have to actually solve the problem, type up solution notes, and then decide how hard it is and what that problem teaches. This might take anywhere from 30 minutes to the entire afternoon, per problem. Now imagine you need 150 such problems to run a year’s curriculum, and you can see why the first year was so stressful. (I was very fortunate to have paid much of this cost in high school; I still remember many of the problems I did back as a student.)

So it seemed like a waste if I spent a lot of time vetting a problem and then my students didn’t do it, and as practical matter I didn’t have enough materials yet to have much leeway anyways. I told myself this would be fine: after all, if you couldn’t do a problem, all you had to do was tell me what you’ve tried, and then I’d walk you through the rest of it. So there’s no reason why you couldn’t finish the problem sets, right? (Ha. Ha. Ha.)

Now my problem bank has gotten much deeper, so I don’t have that excuse anymore. [1]

## 3. Agonizing over problem eight

But I’ll tell you now that even before I decided to switch to points, one of the biggest headaches was always whether to add in that an eighth problem that was really nice but also difficult. (When I first started teaching, my problem sets were typically seven problems long.) If you looked at the TeX source for some of my old handouts, you’ll see lots of problems commented out with a line saying “too long already”.

Teaching OTIS made me appreciate the amount of power I have on the other side of a mentor-student relationship. Basically, when I design a problem set, I am making decisions on behalf of the student: “these are the problems that I think you should work on”. Since my kids are all great students that respect me a lot, they will basically do whatever I tell them to.

That means I used to spend many hours agonizing over that eighth problem or whether to punt it. Yes, they’ll learn a lot if they solve (or don’t solve) it, but it will also take them another two or three hours on top of everything else they’re already doing (OTIS, school, trumpet, track, dance, social, blah blah blah). Is it worth those extra hours? Is it not? I’ve lost sleep over whether I made the right choice on the nights I ended up adding that last hard problem.

But in hindsight the right answer all along was to just let the students decide for themselves, because unlike your average high-school math teacher in a room of decked-out slackers, I have the best students in the world.

## 4. The morning I changed my mind

As I got a deeper database this year and commented more problems out, I started thinking about point-based problem sets. But I can tell you the exact moment when I decided to switch.

On the morning of Sunday November 5, I had a traditional problem set on my desk next to a point-based one. In both cases I had figured out how to do about half the problems required. I noticed that the way the half-full glass of water looked was quite different between them. In the first case, I was freaking out about the other half of the problems I hadn’t solved yet. In the second case, I was trying to decide which of the problems would be the most fun to do next.

Then I realized that OTIS was running on the traditional system, and what I had been doing to my students all semester! So instead of doing either problem set I began the first prototypes of the points system.

## 5. Count up

I’m worried I’ll get misinterpreted as arguing that students shouldn’t work hard. This is not really the point. If you read the specification at the beginning carefully, the number of problems the students are solving is actually roughly the same in both systems.

It might be more psychological than anything else: I want my kids to count how many problems they’ve solved, not how many problems they haven’t solved. Every problem you solve makes you better. Every problem you try and don’t solve makes you better, too. But a problem you didn’t have time to try doesn’t make you worse.

I’ll admit to being mildly pissed off at high school for having built this particular mindset into all my kids. The straight-A students sitting in calculus BC aren’t counting how many questions they’ve answered correctly when checking grades. They’re counting how many points they lost. The implicit message is that if you don’t do nearly all the questions, you’re a bad person because you didn’t try hard enough and you won’t learn anything this way and shame on you and…

That can’t possibly be correct. Imagine two calculus teachers A and B using the same textbook. Teacher A assigns 15 questions of homework a week, teacher B assigns 25 questions. All of teacher A’s students are failing by B’s standards. Fortunately, that’s not actually how the world works.

For this reason I’m glad that all the olympiad kids report their performance as “I solved problems 1,2,4,5” rather than “I missed problems 3,6”.

## 6. There are no stupid or lazy questions

The other wrong assumption I had about traditional problem sets was the bit about asking for help on problems you can’t solve. It turns out getting students to ask for help is a struggle. So one other hope is that with the point-based system is that if a student tries a problem, can’t solve it, and is too shy to ask, then they can switch to a different problem and read the solution later on. No need to get me involved with every single missed problem any more.

But anyways I have a hypothesis why asking for help seems so hard (though there are probably other reasons too).

You’ve all heard the teachers who remind students to always ask questions during lectures [2], because it means someone else has the same question. In other words: don’t be afraid to ask questions just because you’re afraid you’ll look dumb, because “there are no stupid questions“.

But I’ve rarely heard anyone say the same thing about problem sets.

As I’m writing this, I realize that this is actually the reason I’ve never been willing to go to office hours to ask my math professors for help on homework problems I’m stuck on. It’s not because I’m worried my professors will think I’m dumb. It’s because I’m worried they’ll think I didn’t try hard enough before I gave up and came to them for help, or even worse, that I just care about my grade. You’ve all heard the freshman biology TA’s complain about those kids that just come and ask them to check all their pset answers one by one, or that come to argue about points they got docked, or what-have-you. I didn’t want to be that guy.

Maybe this shaming is intentional if the class you’re teaching is full of slackers that don’t work unless you crack the whip. [3] But if you are teaching a math class that’s half MOPpers, I seriously don’t think we need guilt-trips for these kids whenever they can’t solve a USAMO3.

So for all my students, here’s my version of the message: there are no stupid questions, and there are no lazy questions.

### Footnotes

1. The other reason I used traditional problem sets at first was that I wanted to force the students to at least try the harder problems. This is actually my main remaining concern about switching to point-based problem sets: you could in principle always ignore the 9-point problems at the end. I tried to compensate for this by either marking some 9’s in red, or else making it difficult to reach the goal without solving at least one 9. I’m not sure this is enough.
2. But if my question is “I zoned out for the last five minutes because I was responding to my friends on snapchat, what just happened?”, I don’t think most professors would take too kindly. So it’s not true literally all questions are welcome in lectures.
3. As an example, the 3.091 class policies document includes FAQ such as “that sounds like a lot of work, is there a shortcut?”, “but what do I need to learn to pass the tests?”, and “but I just want to pass the tests…”. Also an entire paragraph explaining why skipping the final exam makes you a terrible person, including reasons such as “how do you anything is how you do everything”, “students earning A’s are invited to apply as tutors/graders”, and “in college it’s up to you to take responsibility for your academic career”, and so on ad nauseum.

In a previous post I tried to make the point that math olympiads should not be judged by their relevance to research mathematics. In doing so I failed to actually explain why I think math olympiads are a valuable experience for high schoolers, so I want to make amends here.

## 1. Summary

In high school I used to think that math contests were primarily meant to encourage contestants to study some math that is (much) more interesting than what’s typically shown in high school. While I still think this is one goal, and maybe it still is the primary goal in some people’s minds, I no longer believe this is the primary benefit.

My current belief is that there are two major benefits from math competitions:

1. To build a social network for gifted high school students with similar interests.
2. To provide a challenging experience that lets gifted students grow and develop intellectually.

I should at once disclaim that I do not claim these are the only purpose of mathematical olympiads. Indeed, mathematics is a beautiful subject and introducing competitors to this field of study is of course a great thing (in particular it was life-changing for me). But as I have said before, many alumni of math olympiads do not eventually become mathematicians, and so in my mind I would like to make the case that these alumni have gained a lot from the experience anyways.

## 2. Social experience

Now that we have email, Facebook, Art of Problem Solving, and whatnot, the math contest community is much larger and stronger than it’s ever been in the past. For the first time, it’s really possible to stay connected with other competitors throughout the entire year, rather than just seeing each other a handful of times during contest season. There’s literally group chats of contestants all over the country where people talk about math problems or the solar eclipse or share funny pictures or inside jokes or everything else. In many ways, being part of the high school math contest community is a lot like having access to the peer group at a top-tier university, except four years earlier.

There’s some concern that a competitive culture is unhealthy for the contestants. I want to make a brief defense here.

I really do think that the contest community is good at being collaborative rather than competitive. You can imagine a world where the competitors think about contests in terms of trying to get a better score than the other person. [1] That would not be a good world. But I think by and large the community is good at thinking about it as just trying to maximize their own score. The score of the person next to you isn’t supposed to matter (and thinking about it doesn’t help, anyways).

Put more bluntly, on contest day, you have one job: get full marks. [2]

Because we have a culture of this shape, we now get a group of talented students all working towards the same thing, rather than against one another. That’s what makes it possible to have a self-supportive community, and what makes it possible for the contestants to really become friends with each other.

I think the strongest contestants don’t even care about the results of contests other than the few really important ones (like USAMO/IMO). It is a long-running joke that the Harvard-MIT Math Tournament is secretly just a MOP reunion, and I personally see to it that this happens every year. [3]

I’ve also heard similar sentiments about ARML:

I enjoy ARML primarily based on the social part of the contest, and many people agree with me; the highlight of ARML for some people is the long bus ride to the contest. Indeed, I think of ARML primarily as a social event, with some mathematics to make it look like the participants are actually doing something important.

(Don’t tell the parents.)

## 3. Intellectual growth

My view is that if you spend a lot of time thinking or working about anything deep, then you will learn and grow from the experience, almost regardless of what that thing is at an object level. Take chess as an example — even though chess definitely has even fewer “real-life applications” than math, if you take anyone with a 2000+ rating I don’t think many of them would think that the time they invested into the game was wasted. [4]

Olympiad mathematics seems to be no exception to this. In fact the sheer depth and difficulty of the subject probably makes it a particularly good example. [5]

I’m now going to fill this section with a bunch of examples although I don’t claim the list is exhaustive. First, here are the ones that everyone talks about and more or less agrees on:

• Learning how to think, because, well, that’s how you solve a contest problem.
• Learning to work hard and not give up, because the contest is difficult and you will not win by accident; you need to actually go through a lot of training.
• Dual to above, learning to give up on a problem, because sometime the problem really is too hard for you and you won’t solve it even if you spend another ten or twenty or fifty hours, and you have to learn to cut your losses. There is a balancing act here that I think really is best taught by experience, rather than the standard high-school moral cheerleading where you are supposed to “never give up” or something.
• But also learning to be humble or to ask for help, which is a really hard thing for a lot of young contestants to do.
• Learning to be patient, not only with solving problems but with the entire journey. You usually do not improve dramatically overnight.

Here are some others I also believe, but don’t hear as often.

• Learning to be independent, because odds are your high-school math teacher won’t be able to help you with USAMO problems. Training for the highest level of contests is these days almost always done more or less independently. I think having the self-motivation to do the training yourself, as well as the capacity to essentially have to design your own training (making judgments on what to work on, et cetera) is itself a valuable cross-domain skill. (I’m a little sad sometimes that by teaching I deprive my students of the opportunity to practice this. It is a cost.)
• Being able to work neatly, not because your parents told you to but because if you are sloppy then it will cost you points when you make small (or large) errors on IMO #1. Olympiad problems are difficult enough as is, and you do not want to let them become any harder because of your own sloppiness. (And there are definitely examples of olympiad problems which are impossible to solve if you are not organized.)
• Being able to organize and write your thoughts well, because some olympiad problems are complex and requires putting together more than one lemma or idea together to solve. For this to work, you need to have the skill of putting together a lot of moving parts into a single coherent argument. Bonus points here if your audience is someone you care about (as opposed to a grader), because then you have to also worry about making the presentation as clean and natural as possible.

These days, whenever I solve a problem I always take the time to write it up cleanly, because in the process of doing so I nearly always find ways that the solution can be made shorter or more elegant, or at least philosophically more natural. (I also often find my solution is wrong.) So it seems that the write-up process here is not merely about presenting the same math in different ways: the underlying math really does change. [6]

• Thinking about how to learn. For example, the Art of Problem Solving forums are often filled with questions of the form “what should I do?”. Many older users find these questions obnoxious, but I find them desirable. I think being able to spend time pondering about what makes people improve or learn well is a good trait to develop, rather than mindlessly doing one book after another.

Of course, many of the questions I referred to are poor, either with no real specific direction: often the questions are essentially “what book should I read?”, or “give me a exhaustive list of everything I should know”. But I think this is inevitable because these are people’s first attempts at understanding contest training. Just like the first difficult math contest you take often goes quite badly, the first time you try to think about learning, you will probably ask questions you will be embarrassed about in five years. My hope is that as these younger users get older and wiser, the questions and thoughts become mature as well. To this end I do not mind seeing people wobble on their first steps.

• Being honest with your own understanding, particularly of fundamentals. When watching experienced contestants, you often see people solving problems using advanced techniques like Brianchon’s theorem or the n-1 equal value principle or whatever. It’s tempting to think that if you learn the names and statements of all these advanced techniques then you’ll be able to apply them too. But the reality is that these techniques are advanced for a reason: they are hard to use without mastery of fundamentals.

This is something I definitely struggled with as a contestant: being forced to patiently learn all the fundamentals and not worry about the fancy stuff. To give an example, the 2011 JMO featured an inequality which was routine for experienced or well-trained contestants, but “almost impossible for people who either have not seen inequalities at all or just like to compile famous names in their proofs”. I was in the latter category, and tried to make up a solution using multivariable Jensen, whatever that meant. Only when I was older did I really understand what I was missing.

• Dual to the above, once you begin to master something completely you start to learn what different depths of understanding feel like, and an appreciation for just how much effort goes into developing a mastery of something.
• Being able to think about things which are not well-defined. This one often comes as a surprise to people, since math is a field which is known for its precision. But I still maintain that this a skill contests train for.

A very simple example is a question like, “when should I use the probabilistic method?”. Yes, we know it’s good for existence questions, but can we say anything more about when we expect it to work? Well, one heuristic (not the only one) is “if a monkey could find it” — the idea that a randomly selected object “should” work. But obviously something like this can’t be subject to a (useful) formal definition that works 100% of the time, and there are plenty of contexts in which even informally this heuristic gives the wrong answer. So that’s an example of a vague and nebulous concept that’s nonetheless necessary in order to understanding the probabilistic method well.

There are much more general examples one can say. What does it mean for a problem to “feel projective”? I can’t tell you a hard set of rules; you’ll have to do a bunch of examples and gain the intuition yourself. Why do I say this problem is “rigid”? Same answer. How do you tell which parts of this problem are natural, and which are artificial? How do you react if you have the feeling the problem gives you nothing to work with? How can you tell if you are making progress on a problem? Trying to figure out partial answers to these questions, even if they can’t be put in words, will go a long way in improving the mythical intuition that everyone knows is so important.

It might not be unreasonable to say that by this point we are studying philosophy, and that’s exactly what I intend. When I teach now I often make a point of referring to the “morally correct” way of thinking about things, or making a point of explaining why X should be true, rather than just providing a proof. I find this type of philosophy interesting in its own right, but that is not the main reason I incorporate it into my teaching. I teach the philosophy now because it is necessary, because you will solve fewer problems without that understanding.

## 4. I think if you don’t do well, it’s better to you

But I think the most surprising benefit of math contests is that most participants won’t win. In high school everyone tells you that if you work hard you will succeed. The USAMO is a fantastic counterexample to this. Every year, there are exactly 12 winners on the USAMO. I can promise you there are far more than 12 people who work very hard every year with the hope of doing well on the USAMO. Some people think this is discouraging, but I find it desirable.

Let me tell you a story.

Back in September of 2015, I sneaked in to the parent’s talk at Math Prize for Girls, because Zuming Feng was speaking and I wanted to hear what he had to say. (The whole talk was is available on YouTube now.) The talk had a lot of different parts that I liked, but one of them struck me in particular, when he recounted something he said to one of his top students:

I really want you to work hard, but I really think if you don’t do well, if you fail, it’s better to you.

I had a hard time relating to this when I first heard it, but it makes sense if you think about it. What I’ve tried to argue is that the benefit of math contests is not that the contestant can now solve N problems on USAMO in late April, but what you gain from the entire year of practice. And so if you hold the other 363 days fixed, and then vary only the final outcome of the USAMO, which of success and failure is going to help a contestant develop more as a person?

For that reason I really like to think that the final lesson from high school olympiads is how to appreciate the entire journey, even in spite of the eventual outcome.

### Footnotes

1. I actually think this is one of the good arguments in favor of the new JMO/USAMO system introduced in 2010. Before this, it was not uncommon for participants in 9th and 10th grade to really only aim for solving one or two entry-level USAMO problems to qualify for MOP. To this end I think the mentality of “the cutoff will probably only be X, so give up on solving problem six” is sub-optimal.
2. That’s a Zuming quote.
3. Which is why I think the HMIC is actually sort of pointless from a contestant’s perspective, but it’s good logistics training for the tournament directors.
4. I could be wrong about people thinking chess is a good experience, given that I don’t actually have any serious chess experience beyond knowing how the pieces move. A cursory scan of the Internet suggests otherwise (was surprised to find that Ben Franklin has an opinion on this) but it’s possible there are people who think chess is a waste of time, and are merely not as vocal as the people who think math contests are a waste of time.
5. Relative to what many high school students work on, not compared to research or something.
6. Privately, I think that working in math olympiads taught me way more about writing well than English class ever did; English class always felt to me like the skill of trying to sound like I was saying something substantial, even when I wasn’t.

(Ed Note: This was earlier posted under the incorrect title “On Designing Olympiad Training”. How I managed to mess that up is a long story involving some incompetence with Python scripts, but this is fixed now.)

Spoiler warnings: USAMO 2014/1, and hints for Putnam 2014 A4 and B2. You may want to work on these problems yourself before reading this post.

## 1. An Apology

At last year’s USA IMO training camp, I prepared a handout on writing/style for the students at MOP. One of the things I talked about was the “ocean-crossing point”, which for our purposes you can think of as the discrete jump from a problem being “essentially not solved” (${0+}$) to “essentially solved” (${7-}$). The name comes from a Scott Aaronson post:

Suppose your friend in Boston blindfolded you, drove you around for twenty minutes, then took the blindfold off and claimed you were now in Beijing. Yes, you do see Chinese signs and pagoda roofs, and no, you can’t immediately disprove him — but based on your knowledge of both cars and geography, isn’t it more likely you’re just in Chinatown? . . . We start in Boston, we end up in Beijing, and at no point is anything resembling an ocean ever crossed.

I then gave two examples of how to write a solution to the following example problem.

Problem 1 (USAMO 2014)

Let ${a}$, ${b}$, ${c}$, ${d}$ be real numbers such that ${b-d \ge 5}$ and all zeros ${x_1}$, ${x_2}$, ${x_3}$, and ${x_4}$ of the polynomial ${P(x)=x^4+ax^3+bx^2+cx+d}$ are real. Find the smallest value the product

$\displaystyle (x_1^2+1)(x_2^2+1)(x_3^2+1)(x_4^2+1)$

can take.

Proof: (Not-so-good write-up) Since ${x_j^2+1 = (x+i)(x-i)}$ for every ${j=1,2,3,4}$ (where ${i=\sqrt{-1}}$), we get ${\prod_{j=1}^4 (x_j^2+1) = \prod_{j=1}^4 (x_j+i)(x_j-i) = P(i)P(-i)}$ which equals to ${|P(i)|^2 = (b-d-1)^2 + (a-c)^2}$. If ${x_1 = x_2 = x_3 = x_4 = 1}$ this is ${16}$ and ${b-d = 5}$. Also, ${b-d \ge 5}$, this is ${\ge 16}$. $\Box$

Proof: (Better write-up) The answer is ${16}$. This can be achieved by taking ${x_1 = x_2 = x_3 = x_4 = 1}$, whence the product is ${2^4 = 16}$, and ${b-d = 5}$.

Now, we prove this is a lower bound. Let ${i = \sqrt{-1}}$. The key observation is that

$\displaystyle \prod_{j=1}^4 \left( x_j^2 + 1 \right) = \prod_{j=1}^4 (x_j - i)(x_j + i) = P(i)P(-i).$

Consequently, we have

\displaystyle \begin{aligned} \left( x_1^2 + 1 \right) \left( x_2^2 + 1 \right) \left( x_3^2 + 1 \right) \left( x_1^2 + 1 \right) &= (b-d-1)^2 + (a-c)^2 \\ &\ge (5-1)^2 + 0^2 = 16. \end{aligned}

This proves the lower bound. $\Box$

You’ll notice that it’s much easier to see the key idea in the second solution: namely,

$\displaystyle \prod_j (x_j^2+1) = P(i)P(-i) = (b-d-1)^2 + (a-c)^2$

which allows you use the enigmatic condition ${b-d \ge 5}$.

Unfortunately I have the following confession to make:

In practice, most solutions are written more like the first one than the second one.

The truth is that writing up solutions is sort of a chore that people never really want to do but have to — much like washing dishes. So must solutions won’t be written in a way that helps you learn from them. This means that when you read solutions, you should assume that the thing you really want (i.e., the ocean-crossing point) is buried somewhere amidst a haystack of other unimportant details.

## 2. Diff

But in practice even the “better write-up” I mentioned above still has too much information in it.

Suppose you were explaining how to solve this problem to a friend. You would probably not start your explanation by saying that the minimum is ${16}$, achieved by ${x_1 = x_2 = x_3 = x_4 = 1}$ — even though this is indeed a logically necessary part of the solution. Instead, the first thing you would probably tell them is to notice that

$\displaystyle \prod_{j=1}^4 \left( x_j^2 + 1 \right) = P(i)P(-i) = (b-d-1)^2 + (a-c)^2 \ge 4^2 = 16.$

In fact, if your friend has been working on the problem for more than ten minutes, this is probably the only thing you need to tell them. They probably already figured out by themselves that there was a good chance the answer would be ${2^4 = 16}$, just based on the condition ${b-d \ge 5}$. This “one-liner” is all that they need to finish the problem. You don’t need to spell out to them the rest of the details.

When you explain a problem to a friend in this way, you’re communicating just the difference: the one or two sentences such that your friend could work out the rest of the details themselves with these directions. When reading the solution yourself, you should try to extract the main idea in the same way. Olympiad problems generally have only a few main ideas in them, from which the rest of the details can be derived. So reading the solution should feel much like searching for a needle in a haystack.

## 3. Don’t Read Line by Line

In particular: you should rarely read most of the words in the solution, and you should almost never read every word of the solution.

Whenever I read solutions to problems I didn’t solve, I often read less than 10% of the words in the solution. Instead I search aggressively for the one or two sentences which tell me the key step that I couldn’t find myself. (Functional equations are the glaring exception to this rule, since in these problems there sometimes isn’t any main idea other than “stumble around randomly”, and the steps really are all about equally important. But this is rarer than you might guess.)

I think a common mistake students make is to treat the solution as a sequence of logical steps: that is, reading the solution line by line, and then verifying that each line follows from the previous ones. This seems to entirely miss the point, because not all lines are created equal, and most lines can be easily derived once you figure out the main idea.

If you find that the only way that you can understand the solution is reading it step by step, then the problem may simply be too hard for you. This is because what counts as “details” and “main ideas” are relative to the absolute difficulty of the problem. Here’s an example of what I mean: the solution to a USAMO 3/6 level geometry problem, call it ${P}$, might look as follows.

Proof: First, we prove lemma ${L_1}$. (Proof of ${L_1}$, which is USAMO 1/4 level.)

Then, we prove lemma ${L_2}$. (Proof of ${L_2}$, which is USAMO 1/4 level.)

Finally, we remark that putting together ${L_1}$ and ${L_2}$ solves the problem. $\Box$

Likely the main difficulty of ${P}$ is actually finding ${L_1}$ and ${L_2}$. So a very experienced student might think of the sub-proofs ${L_i}$ as “easy details”. But younger students might find ${L_i}$ challenging in their own right, and be unable to solve the problem even after being told what the lemmas are: which is why it is hard for them to tell that ${\{L_1, L_2\}}$ were the main ideas to begin with. In that case, the problem ${P}$ is probably way over their head.

This is also why it doesn’t make sense to read solutions to problems which you have not worked on at all — there are often details, natural steps and notation, et cetera which are obvious to you if and only if you have actually tried the problem for a little while yourself.

## 4. Reflection

The earlier sections describe how to extract the main idea of an olympiad solution. This is neat because instead of having to remember an entire solution, you only need to remember a few sentences now, and it gives you a good understanding of the solution at hand.

But this still isn’t achieving your ultimate goal in learning: you are trying to maximize your scores on future problems. Unless you are extremely fortunate, you will probably never see the exact same problem on an exam again.

So one question you should often ask is:

“How could I have thought of that?”

(Or in my case, “how could I train a student to think of this?”.)

There are probably some surface-level skills that you can pick out of this. The lowest hanging fruit is things that are technical. A small number of examples, with varying amounts of depth:

• This problem is “purely projective”, so we can take a projective transformation!
• This problem had a segment ${AB}$ with midpoint ${M}$, and a line ${\ell}$ parallel to ${AB}$, so I should consider projecting ${(AB;M\infty)}$ through a point on ${\ell}$.
• Drawing a grid of primes is the only real idea in this problem, and the rest of it is just calculations.
• This main claim is easy to guess since in some small cases, the frogs have “violating points” in a large circle.
• In this problem there are ${n}$ numbers on a circle, ${n}$ odd. The counterexamples for ${n}$ even alternate up and down, which motivates proving that no three consecutive numbers are in sorted order.
• This is a juggling problem!

(Brownie points if any contest enthusiasts can figure out which problems I’m talking about in this list!)

## 5. Learn Philosophy, not Formalism

But now I want to point out that the best answers to the above question are often not formalizable. Lists of triggers and actions are “cheap forms of understanding”, because going through a list of methods will only get so far.

On the other hand, the un-formalizable philosophy that you can extract from reading a question, is part of that legendary “intuition” that people are always talking about: you can’t describe it in words, but it’s certainly there. Maybe I would even be better if I reframed the question as:

“What does this problem feel like?”

So let’s talk about our feelings. Here is David Yang’s take on it:

Whenever you see a problem you really like, store it (and the solution) in your mind like a cherished memory . . . The point of this is that you will see problems which will remind you of that problem despite having no obvious relation. You will not be able to say concretely what the relation is, but think a lot about it and give a name to the common aspect of the two problems. Eventually, you will see new problems for which you feel like could also be described by that name.

Do this enough, and you will have a very powerful intuition that cannot be described easily concretely (and in particular, that nobody else will have).

This itself doesn’t make sense without an example, so here is an example of one philosophy I’ve developed. Here are two problems on Putnam 2014:

Problem 2 (Putnam 2014 A4)

Suppose ${X}$ is a random variable that takes on only nonnegative integer values, with ${\mathbb E[X] = 1}$, ${\mathbb E[X^2] = 2}$, and ${\mathbb E[X^3] = 5}$. Determine the smallest possible value of the probability of the event ${X=0}$.

Problem 3 (Putnam 2014 B2)

Suppose that ${f}$ is a function on the interval ${[1,3]}$ such that ${-1\le f(x)\le 1}$ for all ${x}$ and

$\displaystyle \int_1^3 f(x) \; dx=0.$

How large can ${\int_1^3 \frac{f(x)}{x} \; dx}$ be?

At a glance there seems to be nearly no connection between these problems. One of them is a combinatorics/algebra question, and the other is an integral. Moreover, if you read the official solutions or even my own write-ups, you will find very little in common joining them.

Yet it turns out that these two problems do have something in common to me, which I’ll try to describe below. My thought process in solving either question went as follows:

In both problems, I was able to quickly make a good guess as to what the optimal ${X}$/${f}$ was, and then come up with a heuristic explanation (not a proof) why that guess had to be correct, namely, “by smoothing, you should put all the weight on the left”. Let me call this optimal argument ${A}$.

That conjectured ${A}$ gave a numerical answer to the actual problem: but for both of these problems, it turns out that numerical answer is completely uninteresting, as are the exact details of ${A}$. It should be philosophically be interpreted as “this is the number that happens to pop out when you plug in the optimal choice”. And indeed that’s what both solutions feel like. These solutions don’t actually care what the exact values of ${A}$ are, they only care about the properties that made me think they were optimal in the first place.

I gave this philosophy the name Equality, with poster description “problems where looking at the equality case is important”. This text description feels more or less useless to me; I suppose it’s the thought that counts. But ever since I came up with this name, it has helped me solve new problems that come up, because they would give me the same feeling that these two problems did.

Two more examples of these themes that I’ve come up with are Global and Rigid, which will be described in a future post on how I design training materials.

# Against Perfect Scores

One of the pieces of advice I constantly give to young students preparing for math contests is that they should probably do harder problems. But perhaps I don’t preach this zealously enough for them to listen, so here’s a concrete reason (with actual math!) why I give this advice.

## 1. The AIME and USAMO

In the USA many students who seriously prepare for math contests eventually qualify for an exam called the AIME (American Invitational Math Exam). This is a 3-hour exam with 15 short-answer problems; the median score is maybe about 5 problems.

Correctly solving maybe 10 of the problems qualifies for the much more difficult USAMO. This national olympiad is much more daunting, with six proof-based problems given over nine hours. It is not uncommon for olympiad contestants to not solve a single problem (this certainly happened to me a fair share of times!).

You’ll notice the stark difference in the scale of these contests (Tanya Khovanova has a longer complaint about this here). For students who are qualifying for USAMO for the first time, the olympiad is terrifying: I certainly remember the first time I took the olympiad with a super lofty goal of solving any problem.

Now, my personal opinion is that the difference between AIME and USAMO is generally exaggerated, and less drastic than appearances suggest. But even then, the psychological fear is still there — so what do you think happens to this demographic of students?

Answer: they don’t move on from AIME training. They think, “oh, the USAMO is too hard, I can only solve 10 problems on the AIME so I should stick to solving hard problems on the AIME until I can comfortably solve most of them”. So they keep on working through old AIME papers.

## 2. Perfect Scores

To understand why this is a bad idea, let’s ask the following question: how good to you have to be to consistently get a perfect score on the AIME?

Consider first a student averages a score of ${10}$ on the AIME, which is a fairly comfortable qualifying score. For illustration, let’s crudely simplify and assume that on a 15-question exam, he has a independent ${\frac23}$ probability of getting each question right. Then the chance he sweeps the AIME is

$\displaystyle \left( \frac23 \right)^{15} \approx 0.228\%.$

This is pretty low, which makes sense: ${10}$ and ${15}$ on the AIME feel like quite different scores.

Now suppose we bump that up to averaging ${12}$ problems on the AIME, which is almost certainly enough to qualify for the USAMO. This time, the chance of sweeping is

$\displaystyle \left( \frac{4}{5} \right)^{15} \approx 3.52\%.$

This should feel kind of low to you as well. So if you consistently solve ${80\%}$ of problems in training, your chance at netting a perfect score is still dismal, even though on average you’re only three problems away.

Well, that’s annoying, so let’s push this as far as we can: consider a student who’s averaging ${14}$ problems (thus, ${93\%}$ success), id est a near-perfect score. Then the probability of getting a perfect score

$\displaystyle \left( \frac{14}{15} \right)^{15} \approx 35.5\%.$

Which is\dots just over ${\frac 13}$.

At which point you throw up your hands and say, what more could you ask for? I’m already averaging one less than a perfect score, and I still don’t have a good chance of acing the exam? This should feel very unfair: on average you’re only one problem away from full marks, and yet doing one problem better than normal is still a splotchy hit-or-miss.

## 3. Some Combinatorics

Those of you who either know statistics / combinatorics might be able to see what’s going on now. The problem is that

$\displaystyle (1-\varepsilon)^{15} \approx 1 - 15\varepsilon$

for small ${\varepsilon}$. That is, if your accuracy is even a little ${\varepsilon}$ away from perfect, that difference gets amplified by a factor of ${15}$ against you.

Below is a nice chart that shows you, based on this oversimplified naïve model, how likely you are to do a little better than your average.

$\displaystyle \begin{array}{lrrrrrr} \textbf{Avg} & \ge 10 & \ge 11 & \ge 12 & \ge 13 & \ge 14 & \ge 15 \\ \hline \mathbf{10} & 61.84\% & 40.41\% & 20.92\% & 7.94\% & 1.94\% & 0.23\% \\ \mathbf{11} & & 63.04\% & 40.27\% & 19.40\% & 6.16\% & 0.95\% \\ \mathbf{12} & & & 64.82\% & 39.80\% & 16.71\% & 3.52\% \\ \mathbf{13} & & & & 67.71\% & 38.66\% & 11.69\% \\ \mathbf{14} & & & & & 73.59\% & 35.53\% \\ \mathbf{15} & & & & & & 100.00\% \\ \end{array}$

Even if you’re not aiming for that lofty perfect score, we see the same repulsion effect: it’s quite hard to do even a little better than average. If you get an average score of ${k}$, the probability of getting ${k+1}$ looks to be about ${\frac25}$. As for ${k+2}$ the chances are even more dismal. In fact, merely staying afloat (getting at least your average score) isn’t a comfortable proposition.

And this is in my simplified model of “independent events”. Those of you who actually take the AIME know just how costly small arithmetic errors are, and just how steep the difficulty curve on this exam is.

All of this goes to show: to reliably and consistently ace the AIME, it’s not enough to be able to do 95% of AIME problems (which is already quite a feat). You almost need to be able to solve AIME problems in your sleep. On any given AIME some people will get luckier than others, but coming out with a perfect score every time is a huge undertaking.

## 4. 90% Confidence?

By the way, did I ever mention that it’s really hard to be 90% confident in something? In most contexts, 90% is a really big number.

If you don’t know what I’m talking about:

This is also the first page of this worksheet. The idea of this quiz is to give you a sense of just how high 90% is. To do this, you are asked 10 numerical questions and must provide an interval which you think the answer lies within with probability 90%. (So ideally, you would get exactly 9 intervals correct.)

As a hint: almost everyone is overconfident. Second hint: almost everyone is overconfident even after being told that their intervals should be embarrassingly wide. Third hint: I just tried this again and got a low score.

(For more fun of this form: calibration game.)

## 5. Practice

So what do you do if you really want to get a perfect score on the AIME?

Well, first of all, my advice is that you have better things to do (like USAMO). But even if you are unshakeable on your desire to get a 15, my advice still remains the same: do some USAMO problems.

Why? The reason is that going from average ${14}$ to average ${15}$ means going from 95% accuracy to 99% accuracy, as I’ve discussed above.

So what you don’t want to do is keep doing AIME problems. You are not using your time well if you get 95% accuracy in training. I’m well on record saying that you learn the most from problems that are just a little above your ability level, and massing AIME problems is basically the exact opposite of that. You’d maybe only run into a problem you couldn’t solve once every 10 or 20 or 30 problems. That’s just grossly inefficient.

The way out of this is to do harder problems, and that’s why I explicitly suggest people start working on USAMO problems even before they’re 90% confident they will qualify for it. At the very least, you certainly won’t be bored.

I know some friends who are fantastic at synthetic geometry. I can give them any problem and they’ll come up with an incredibly impressive synthetic solution. I also have some friends who are very bad at synthetic geometry, but have such good fortitude at computations that they can get away with using Cartesian coordinates for everything.

I don’t consider myself either of these types; I don’t have much ingenuity when it comes to my solutions, and I’m actually quite clumsy when it comes to long calculations. But nonetheless I have a high success rate with olympiad geometry problems. Not only that, but my solutions are often very algorithmic, in the sense that any well-trained student should be able to come up with this solution.

In this article I try to describe how I come up which such solutions.

## 1. The Three Reductions

Very roughly, there are three different ways I try to make progress on a geometry problem.

• (I) The standard synthetic techniques; angle chasing, cyclic quadrilaterals, homothety, radical axis / power of a point, etc. My own personal arsenal contains some weapons not known to many contestants as well, most notably inversion, harmonic bundles and quadrilaterals, and spiral similarity / Miquel points.For this part, it’s highly advantageous to be well-versed with “standard” configurations and tricks. To give an extreme example: to solve Iran TST 2009, Problem 9 one essentially needs only recognize two configurations: a lemma about the midpoint of an altitude (2002 G7) and another lemma about the line ${EF}$ (USAJMO 2014/6). Not knowing either of these makes it more difficult to solve the problem synthetically in the time limit. As a reference, Yufei Zhao’s lemmas handout contains a fairly comprehensive list of these configurations.

Easier problems don’t require as much in this way of configuration recognition.

• (II) Standard computational techniques (aka bashing). Personally, I prefer complex numbers and barycentric coordinates but I know other students who will use Cartesian coordinates and trigonometry to great success. The advantage of such methods is that they are straightforward and reliable, albeit tedious and time-consuming. It is mostly a matter of experience to understand whether a calculation can be carried out within the time limit — I can basically tell just by looking at a setup whether it can be solved in this time.
• (III) Most surprisingly: simply finding crucial claims. Especially for harder problems like IMO 3/6 much of the time the key to solving a problem is making some key observation. Said another way: a difficult IMO 3/6 problem which asks you to prove ${A \implies B}$ might have a solution which goes like,

$\displaystyle A \implies X \implies Y \implies B.$

Each of the individual implications might be no harder than an IMO 1/4 but the difficulty rests in finding what to prove. The most reliable way to do such things is to draw large, in-scale diagrams. If you are good at recognizing cyclic quadrilaterals, collinear points, etc. then the correct claims will naturally suggest themselves; conversely, good diagrams will prevent you from wasting time trying to prove things that aren’t true (effectively letting you test your claims “experimentally” before trying to prove them).

Type (III) deserves some comment here. There is more to making progress on a problem than simply trying things you think will solve the problem: there is some “scouting” involved that you will need to do for any difficult problems. As a terrible analogy, in StarCraft you have to scout an experienced opponent to understand what they’re doing before you try to attack them. The situation with IMO 3/6 is no different: you have to have some understanding of the problem before you stand a chance of being able to solve it.

Easy problems can often succumb to just one class of attacks, but the interesting and difficult problems can require two or all three classes in order to solve. How much you use each type of strategy is in my opinion a matter of personal taste — some people don’t use (II) at all and rely on (I) to prove everything, and even vice versa! I like to think I balance (I) and (II) evenly. But (III) is indispensable, and in any case I think part of the reason I have been so successful with geometry problems is precisely that I can draw on all three strategies in tandem, rather than being limited to one or two.

In fact, a good rule of thumb that I use for judging the difficulty of a problem is how many of the above methods I had to use: the ${n}$th problem on an IMO paper should require me to resort to about ${n}$ of these strategies.

## 2. Concrete Examples

I’ll now give some concrete examples of the things I said above. Warning: spoilers follow, and hyperlinks lead to my solutions on Art of Problem Solving. You are encouraged to try the problems yourself before reading the comments.

Example [EGMO 2012/1] Let ${ABC}$ be a triangle with circumcenter ${O}$. The points ${D}$, ${E}$, ${F}$ lie in the interiors of the sides ${BC}$, ${CA}$, ${AB}$ respectively, such that ${\overline{DE}}$ is perpendicular to ${\overline{CO}}$ and ${\overline{DF}}$ is perpendicular to ${\overline{BO}}$. Let ${K}$ be the circumcenter of triangle ${AFE}$. Prove that the lines ${\overline{DK}}$ and ${\overline{BC}}$ are perpendicular.

This is a pretty typical entry-level geometry problem. Do some angle chasing (I) to find one cyclic quad (III), and then follow through to solve the problem (I). If you are good enough, you don’t even need to find the cyclic quad in advance; just play around with the angles until you notice it.

Example [IMO 2014, Problem 4] Let ${P}$ and ${Q}$ be on segment ${BC}$ of an acute triangle ${ABC}$ such that ${\angle PAB=\angle BCA}$ and ${\angle CAQ=\angle ABC}$. Let ${M}$ and ${N}$ be the points on ${AP}$ and ${AQ}$, respectively, such that ${P}$ is the midpoint of ${AM}$ and ${Q}$ is the midpoint of ${AN}$. Prove that the intersection of ${BM}$ and ${CN}$ is on the circumference of triangle ${ABC}$.

You can solve this problem by barycentric coordinates (II) instantly (textbook example). Also similar triangles (I) solves the problem pretty quickly as well. Again, this problem is “easy” in the sense that one can directly approach it with either (I) or (II), not needing (III) at all.

Example [USAMO 2015/2] Quadrilateral ${APBQ}$ is inscribed in circle ${\omega}$ with ${\angle P = \angle Q = 90^{\circ}}$ and ${AP = AQ < BP}$. Let ${X}$ be a variable point on segment ${\overline{PQ}}$. Line ${AX}$ meets ${\omega}$ again at ${S}$ (other than ${A}$). Point ${T}$ lies on arc ${AQB}$ of ${\omega}$ such that ${\overline{XT}}$ is perpendicular to ${\overline{AX}}$. Let ${M}$ denote the midpoint of chord ${\overline{ST}}$. As ${X}$ varies on segment ${\overline{PQ}}$, show that ${M}$ moves along a circle.

This was not supposed to be a very difficult problem, but it seems to have nearly swept the JMO group. Essentially, the key to this problem is to notice that the center of the desired circle is in fact the midpoint of ${AO}$ (with ${O}$ the center of the circle). This is a huge example of (III) — after this observation, one can solve the problem very quickly using complex numbers (II). It is much harder (though not impossible) to solve the problem without knowing the desired center.

Example [USAMO 2014/5] Let ${ABC}$ be a triangle with orthocenter ${H}$ and let ${P}$ be the second intersection of the circumcircle of triangle ${AHC}$ with the internal bisector of the angle ${\angle BAC}$. Let ${X}$ be the circumcenter of triangle ${APB}$ and ${Y}$ the orthocenter of triangle ${APC}$. Prove that the length of segment ${XY}$ is equal to the circumradius of triangle ${ABC}$.

Personally I think the most straightforward solution is to use (I) to eliminate the orthocenter condition, and then finish with complex numbers (II). Normally, you won’t see a medium-level problem that dies immediately to (II), and the only reason a problem like this could end up as a problem 5 is that there is a tiny bit of (I) that needs to happen before the complex numbers becomes feasible.

Example [IMO 2014/3] Convex quadrilateral ${ABCD}$ has ${\angle ABC = \angle CDA = 90^{\circ}}$. Point ${H}$ is the foot of the perpendicular from ${A}$ to ${BD}$. Points ${S}$ and ${T}$ lie on sides ${AB}$ and ${AD}$, respectively, such that ${H}$ lies inside triangle ${SCT}$ and

$\displaystyle \angle CHS - \angle CSB = 90^{\circ}, \quad \angle THC - \angle DTC = 90^{\circ}.$

Prove that line ${BD}$ is tangent to the circumcircle of triangle ${TSH}$.

Like most IMO 3/6’s I had to resort to using all three methods in order to solve this problem. The first important step was finding out what to do with the angle condition. It turns out that in fact, it’s equivalent to the circumcenter of triangle ${TCH}$ lying on side ${AD}$ of the triangle (III); proving this is then a matter of angle chasing (I). Afterwards, one has to recognize a tricky usage of the angle bisector theorem (I) to reduce it to something that can be computed with trigonometry (II). This leads to a direct solution that, while not elegant, also requires much less ingenuity then most of the solutions found by friends I know.

I really want to stress that being proficient in all three strategies is key to getting “straightforward” solutions like this to IMO 3/6 caliber problems. If you miss any of these components, you are not going to solve the problem.

Example [IMO 2011/6] Let ${ABC}$ be an acute triangle with circumcircle ${\Gamma}$. Let ${\ell}$ be a tangent line to ${\Gamma}$, and let ${\ell_a}$, ${\ell_b}$, ${\ell_c}$ be the lines obtained by reflecting ${\ell}$ in the lines ${BC}$, ${CA}$, and ${AB}$, respectively. Show that the circumcircle of the triangle determined by the lines ${\ell_a}$, ${\ell_b}$, and ${\ell_c}$ is tangent to the circle ${\Gamma}$.

The ultimate example of these three principles. Using a trick that showed up on APMO 2014/5 and RMM 2013/3, one constructs the tangency point ${T}$ and connects the points ${A_1}$, ${B_1}$, ${C_1}$, as I explain in this post, yielding points ${A_2}$, ${B_2}$, ${C_2}$. After that, a very careful examination of the diagram (possibly several diagrams) leads to a conjecture that ${A_1A=AP}$, et cetera. This is the key observation (III), and leads to highly direct solution via (II). But the point of this problem is that you need to have the guts to construct those auxiliary points and then boldly claim they are the desired “squared” points.

## 3. Comparison with Other Subjects

The approaches I’ve described highlight some of the features of olympiad geometry which distinguish it from other subjects.

• Unlike other olympiad subjects, you can actually obtain a big advantage by just knowing lots of theory. Experienced contestants simply “recognize” a large body of common configurations that those without access to training materials have never seen before. Similarly, there are a lot of fancy techniques that can make a big difference. This is much less true of other subjects (for example combinatorics is the opposite extreme).
• There’s less variance in the subject: lots of Euclidean geometry problems feel the same, and all of them use the same body of techniques. It reminds me of chess: it’s very “narrow” in the sense that at the end of the day, there are only so many possible moves. (Olympiad inequalities also has this kind of behavior.) Again combinatorics is the opposite of this.
• You have a reliable backup in case you can’t find the official solution: bash. Moreover, in general there are often many different ways to solve a problem; not true of other subjects.
• If you want to make some “critical claim” you can quickly test it empirically (by drawing a good diagram).

# What leads to success at math contests?

Updated version of generic advice post: Platitudes v3.

I think this is an important question to answer, not the least of reasons being that understanding how to learn is extremely useful both for teaching and learning. [1]

About a year ago [2], I posted my thoughts on what the most important things were in math contest training. Now that I’m done with the IMO I felt I should probably revisit what I had written.

It looks like the main point of my post a year ago was mainly to debunk the idea that specific resources are important. Someone else phrased this pretty well in the replies to the thread

The issue is many people simply ask about how they should prepare for AIME or USAMO without any real question. They simply figure that AOPS has a lot of successful people that excel at both contests, so why not see what they did? Unfortunately, that’s not how it works – that’s what this post is saying. There is no “right” training.

This is so obvious to me now that I’m going to focus more on what I think actually matters. So I now have the following:

1. Do lots of problems.
2. Learn some standard tricks.
3. Do problems which are just above your reach.
4. Understand the motivation behind solutions to problems you do.
5. Know when to give up.
6. Do lots of problems.

Elaboration on the above:

1. Self-explanatory. I can attest that the Contests section on AoPS suffices.
2. One should, for example, know what a radical axis is. It may also help to know what harmonic quadrilaterals, Karamata, or Kobayashi is, for example, but increasingly obscure things are increasingly less necessary. This step can be achieved by using books/handouts or doing lots of problems.
3. Basically, you improve when you do problems that are hard enough to challenge you but reasonable for you to solve. My rule of thumb is that you shouldn’t be confident that you can solve the practice problem, nor confident that you won’t solve it. There should be suspense.

In my experience, people tend to underestimate themselves — probably my biggest regret was being scared of IMO/USAMO #3’s and #6’s until late in my IMO training, when I finally realized I needed to actually start solving some. I encourage prospective contestants to start earlier.

4. I think the best phrasing of this is, “how would I train a student to be able to solve this problem?”, something I ask myself a lot. By answering this question you also understand

a. Which parts of the solution are main ideas and which steps are routine details,
b. Which parts of the problem are the “hard steps” of the problem,
c. How one would think of the hard steps of the solution,
and so on. I usually like to summarize the hard parts of the solution in a few sentences. As an example, “USAMO 2014 #6 is solved by considering the $N \times N$ grid of primes and noting that small primes cannot cover the board adequately”. Or “ELMO 2013 #5 is solved by considering the 1D case, realizing the answer is $cn^k$, and then generalizing directly to the 3D case”.

In general, after reading a solution, you should be able to state in a couple sentences all the main ideas of the solution, and basically know how to solve the problem from there.

5. In 2011, JMO #5 took me two hours. In 2012, the same problem took me 30 seconds and SL 2011 G4 took me two hours. Today, SL 2011 G4 takes me about five minutes and IMO 2011 #6 took me seven hours. It would not have been a good use of my time in 2011 to spend several hundred hours on IMO #6.

This is in part doing (3) correctly by not doing things way, way over your head and not doing things way below your ability. Regardless you should know when to move on to the next problem. It’s fine to try out really hard problems, just know when more time will not help.

In the other direction, some students give up too early. You should only give up on a problem after you’ve made no progress for a while, and realize you are unlikely to get any further than you already are. My rule of thumb for olympiads is one or two hours without making progress.

6. Self-explanatory.

I think the things I mentioned above are at least extremely useful (“necessary” is harder to argue, but I think you could make a case for it). Now is it sufficient? I have no idea.

##### Footnotes
1. The least of reasons is that people ask me this all the time and I should properly prepare a single generic response.
2. It’s only been a year? I could have sworn it was two or three.