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

# Tag Archives: teaching

# Math contest platitudes, v3

I think it would be nice if every few years I updated my generic answer to “how do I get better at math contests?”. So here is the 2019 version. Unlike previous instances, I’m going to be a little less olympiad-focused than I usually am, since these days I get a lot of people asking for help on the AMC and AIME too.

(Historical notes: you can see the version from right after I graduated and the version from when I was still in high school. I admit both of them make me cringe slightly when I read them today. I still think everything written there is right, but the style and focus seems off to me now.)

## 0. Stop looking for the “right” training (or: be yourself)

These days many of the questions I get are clearly most focused on trying to find a perfect plan — questions like “what did YOU do to get to X” or “how EXACTLY do I practice for Y”. (Often these words are in all-caps in the email, too!) When I see these I always feel very hesitant to answer. The reason is that I always feel like there’s some implicit hope that I can give you some recipe that, if you follow it, will guarantee reaching your goals.

I’m sorry, math contests don’t work that way (and can’t work that way). I actually think that if I gave you a list of which chapters of which books I read in 2009-2010 over which weeks, and which problems I did on each day, and you followed it to the letter, it would go horribly.

Why? It’s not just a talent thing, I think. Solving math problems is actually a deeply personal art: despite being what might appear to be a cold and logical discipline, learning math and getting better at it actually requires being *human*. Different people find different things natural or unnatural, easy or hard, et cetera. If you try to squeeze yourself into some mold or timeline then the results will probably be counterproductive.

On the flip side, this means that you can worry a lot less. I actually think that surprisingly often, you can get a first-order approximation of what’s the “best” thing to do by simply doing whatever feels the most engaging or rewarding (assuming you like math, of course). Of course there are some places where this is not correct (e.g., you might hate geometry, but cannot just ignore it). But the first-order approximation is actually quite decent.

That’s why in the introduction to my geometry book, I explicitly have the line:

Readers are encouraged to not be bureaucratic in their learning and move around as they see fit, e.g., skipping complicated sections and returning to them later, or moving quickly through familiar material.

Put another way: as learning math is quite personal, the advice “**be yourself**” is well-taken.

## 1. Some brief recommendations (anyways)

With all that said, probably no serious harm will come from me listing a little bit of references I think are reasonable — so that you have somewhere to start, and can oscillate from there.

For learning theory and fundamentals:

- AMC – mid AIME: Volume 2
- Late AIME and beyond: E.G.M.O., OTIS Excerpts, and more

For sources of additional practice problems (other than the particular test you’re preparing for):

- The collegiate contests HMMT November, PUMaC, CMIMC will typically have decent short-answer problems.
- HMMT February is by far the hardest short-answer contest I know of.
- At the olympiad level, there are so many national olympiads and team selection tests that you will never finish. (My website has an archive of USA problems and solutions if you’re interested in those in particular.)

The IMO Shortlist is also good place to work as it contains proposals of varying difficulty from many countries — and thus is the most culturally diverse. As for other nations, as a rule of thumb, any country that often finishes in the top 20 at the IMO (say) will probably have a good questions on their national olympiad or TST.

For every subject that’s not olympiad geometry, there are actually surprisingly few named theorems.

## 2. Premature optimization is the root of all evil (so just get your hands dirty)

For some people, the easiest first step to getting better is to double the amount of time you spend practicing. (Unless that amount is zero, in which case, you should just start.)

There is a time and place for spending time thinking about how to practice — one example is if you’ve been working a while and feel like nothing has changed, or you’ve been working on some book and it just doesn’t feel fun, etc. Another common example is if you notice you keep missing all the functional equations on the USAMO: then, maybe it’s time to search up some handouts on functional equations. Put another way, if you feel *stuck*, then you can start thinking about whether you’re not doing something right.

On the other extreme, if you’re wondering whether you are ready to read book X or do problems from Y contest, my advice is to just try it and see if you like it. There is no commitment: just read Chapter 1, see how you feel. If it works, keep doing it, if not, try something else.

(I can draw an analogy from my own life. Whenever I am learning a new board game or card game, like Catan or Splendor or whatever, I always overthink it. I spend all this time thinking and theorizing and trying to come up with this brilliant strategy — which never works, because it’s my *first game*, for crying out loud. It turns out that until you start grappling at close range and getting your hands dirty, your internal model of something you’ve never done is probably not that good.)

## 3. Doing problems just above your level (and a bit on reflecting on them)

There is one pitfall that I do see sometimes, common enough I will point it out. If you mostly (only?) do old practice tests or past problems, then you’re liable to be spending too much time on easy problems. That was the topic of another old post of mine, but the short story is that if you find yourself constantly getting 130ish on AMC10 practice tests, then maybe you should spend most of your time working on problems 21-25 rather than repeatedly grinding 1-20 over and over. (See 28:30-29:00 here to hear Zuming make fun of them.)

The common wisdom is that you should **consistently do problems just above your level** so that you gradually increase the difficulty of problems you are able to solve. The situation is a little more nuanced at the AMC/AIME level, since for those short-answer contests it’s *also* important to be able to do routine problems quickly and accurately. However, I think for most people, you really should be spending at least 70% of your time getting smarter, rather than just faster.

I think in this case, I want to give concrete descriptions. Here’s some examples of what can happen after a problem.

*You looked at the problem and immediately (already?) knew how to do it.*Then you probably didn’t learn much from it. (But at least you’ll get faster, if not smarter.)*You looked at the problem and didn’t know right away how to start, but after a little while figured it out.*That’s a little better.*You struggled with the problem and eventually figured out a solution, but maybe not the most elegant one.*I think that’s a great situation to be in. You came up with*some*solution to the problem, so you understand it fairly well, but there’s still more for you to update your instincts on. What can you do in the future to get solutions more like the elegant one?*You struggled with the problem and eventually gave up, then when you read the solution you realize quickly what you were missing.*I think that’s a great situation to be in, too. You now want to update your instincts by a little bit — how could you make sure you don’t miss something like that again in the future?*The official solution quoted some theorem you don’t know.*If this was among a batch of problems where the other problems felt about the right level to you, then I think often this is a pretty good time to see if you can learn the statement (better, proof) of the theorem. You have just spent some time working on a situation in which the theorem was useful, so that data is fresh in your mind. And pleasantly often, you will find that ideas you came up with during your attempt on the problem correspond to ideas in the statement or proof of the theorem, which is great!*You didn’t solve the problem, and the solution makes sense, but you don’t see how you would have come up with it.*It’s possible that this is the fault of the solutions author (many people are actually quite bad at making solutions read naturally). If you have a teacher, this is the right time to ask them about it. But it’s also possible that the problem was too hard. In general, I think it’s better to miss problems “by a little”, whatever that means, so that you can update your intuition correctly.*You can’t even understand the solution.*Okay, too hard.

You’ll notice how much emphasis I place on the post-problem reflection process. This is actually important — after all the time you spent working on the problem itself, you want to update your instincts as much as possible to get the payoff. In particular, I think it’s usually worth it to read the solutions to problems you worked on, whether or not you solve them. In general, after reading a solution, I think 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.

For the olympiad level, I have a whole different post dedicated to reading solutions, and interested readers can read more there. (One point from that post I do want to emphasize since it wasn’t covered explicitly in any of the above examples: by USA(J)MO level it becomes important to begin building intuition that you can’t explicitly formalize. You may start having vague feelings and notions that you can’t quite put your finger on, but you can feel it. These non-formalizable feelings are valuable, take note of them.)

## 4. Leave your ego out (e.g. be willing to give up on problems)

This is easy advice to give, but it’s hard advice to follow. For concreteness, here are examples of things I think can be explained this way.

Sometimes people will ask me whether they need to solve *every* problem in each chapter of EGMO, or do *every* past practice test, or so on. The answer is: of course not, and why would you even think that? There’s nothing magical about doing 80% of the problems versus 100% of them. (If there was, then EGMO is secretly a terrible book, because I commented out some problems, and so OH NO YOU SKIPPED SOME AAAHHHHH.) And so it’s okay to start Chapter 5 even though you didn’t finish that last challenge problem at the end. Otherwise you let one problem prevent you from working on the next several.

Or, sometimes I learn about people who, if they do not solve an olympiad problem, will refuse to look at the solution; instead they will mark it in a spreadsheet and to come back to later. In short, they *never* give up on a problem: which I think is a bad idea, since reflecting on missed problems is so important. (It is not as if you can realistically run out of olympiad problems to do.) And while this is still better than giving up too early, I mean, all things in moderation, right?

I think if somehow people were able to completely leave your ego out, and not worry at all about how good you are and rather just maximize learning, then mistakes like these two would be a lot rarer. Of course, this is impossible to do in practice (we’re all human), but it’s good to keep in mind at least that this is an ideal we can strive for.

## 5. Enjoy it

Which leads me to the one bit that everyone already knows, but that no platitude-filled post would be complete without: to do well at math contests (or anything hard) you probably have to enjoy the process of getting better. Not just the end result. You have to enjoy the work itself.

Which is not to say you have to do it all the time or for hours a day. Doing math is hard, so you get tired eventually, and beyond that forcing yourself to work is not productive. Thus when I see people talk about how they plan to do every shortlist problem, or they will work N hours per day over M time, I always feel a little uneasy, because it always seems too results-oriented.

In particular, I actually think it’s quite hard to spend more than two or three good hours per day on a regular basis. I certainly never did — back in high school (and even now), if I solved one problem that took me more than an hour, that was considered a good day. (But I should also note that the work ethic of my best students consistently amazes me; it far surpasses mine.) In that sense, the learning process can’t be forced or rushed.

There is one sense in which you *can* get more hours a day, that I am on record saying quite often: if you think about math in the shower, then you know you’re doing it right.

# Make training non zero-sum

Some thoughts about some modern trends in mathematical olympiads that may be concerning.

## I. The story of the barycentric coordinates

I worry about my geometry book. To explain why, let me tell you a story.

When I was in high school about six years ago, barycentric coordinates were nearly unknown as an olympiad technique. I only heard about it from whispers in the wind from friends who had heard of the technique and thought it might be usable. But at the time, there were nowhere where everything was written down explicitly. I had a handful of formulas online, a few helpful friends I can reach out to, and a couple example posts littered across some forums.

Seduced by the possibility of arcane power, I didn’t let this stop me. Over the spring of 2012, spring break settled in, and I spent that entire week developing the entire theory of barycentric coordinates from scratch. There were no proofs I could find online, so I had to personally reconstruct all of them. In addition, I set out to finding as many example problems as I could, but since no one had written barycentric solutions yet, I had to not only identify which problems like they might be good examples but also solve them myself to see if my guesses were correct. I even managed to prove a “new” theorem about perpendicular displacement vectors (which I did not get to name after myself).

I continued working all the way up through the summer, adding several new problems that came my way from MOP 2012. Finally, I posted a rough article with all my notes, examples, and proofs, which you can still find online. I still remember this as a sort of *magnus opus* from the first half of high school; it was an immensely rewarding learning experience.

Today, all this and much more can be yours for just $60, with any major credit or debit card.

Alas, my geometry book is just one example of ways in which the math contest scene is looking more and more like an industry. Over the years, more and more programs dedicated to training for competitions are springing up, and these programs can be quite costly. I myself run a training program now, which is even more expensive (in my defense, it’s one-on-one teaching, rather than a residential camp or group lesson).

It’s possible to imagine a situation in which the contest problems become more and more routine. In that world, math contests become an arms race. It becomes mandatory to have training in increasingly obscure techniques: everything from Popoviciu to Vieta jumping to rectangular circumhyperbolas. Students from less well-off families, or even countries without access to competition resources, become unable to compete, and are pushed to the bottom of the IMO scoreboard.

(Fortunately for me, I found out at the 2017 IMO that my geometry book actually *helped* level the international playing field, contrary to my initial expectations. It’s unfortunate that it’s not free, but it turned out that many students in other countries had until then found it nearly impossible to find suitable geometry materials. So now many more people have access to a reasonable geometry reference, rather than just the top countries with well-established training.)

## II. Another dark future

The first approximation you might have now is that training is bad. But I think that’s the wrong conclusion, since, well, I have an entire previous post dedicated to explaining what I perceive as the benefits of the math contest experience. So I think **the conclusion is not that training is intrinsically bad, but rather than training must be meaningful**. That is, the students have to gain something from the experience that’s not just a +7 bonus on their next olympiad contest.

I think the message “training is bad” might be even more dangerous.

Imagine that the fashion swings the other way. The IMO jury become alarmed at the trend of train-able problems, and in response, the problems become designed specifically to antagonize trained students. The entire Geometry section of the IMO shortlist ceases to exist, because some Asian kid wrote this book that gives you too much of an advantage if you’ve read it, and besides who does geometry after high school anyways? The IMO 2014 used to be notable for having three combinatorics problems, but by 2040 the norm is to have four or five, because everyone knows combinatorics is harder to train for.

Gradually, the IMO is redesigned to become an IQ test.

The changes then begin to permeate down. The USAMO committee is overthrown, and USAMO 2050 features six linguistics questions “so that we can find out who can actually think”. Math contests as a whole become a system for identifying the best genetic talent, explicitly aimed at weeding out the students who have “just been trained”. It doesn’t matter how hard you’ve worked; we want “creativity”.

This might be great at identifying the best mathematicians each generation, but I think an IMO of this shape would be actively destructive towards the contestants and community as well. You thought math contests were bad because they’re discouraging to the kids who don’t win? What if they become redesigned to make sure that *you can’t improve your score no matter how hard you work*?

## III. Now

What this means is that we have a balancing act to maintain. We do not want to eliminate the role of training entirely, because the whole point of math contests is to have a learning experience that lasts longer than the two-day contest every year. But at the same time, we need to ensure the training is interesting, that it is deep and teaches skills like the ones I described before.

Paying $60 to buy a 300-page PDF is not meaningful. But spending many hours to work through the problems in that PDF might be.

In many ways this is not a novel idea. If I am trying to teach a student, and I give them a problem which is too easy, they will not learn anything from it. Conversely, if I give them a problem which is too difficult, they will get discouraged and are unlikely to learn much from their trouble. The situation with olympiad training feels the same.

This applies to the way I think about my teaching as well. I am always upset when I hear (as I have) things like “X only did well on USAMO because of Evan Chen’s class”. If that is true, then all I am doing is taking money as input and changing the results of a zero-sum game as output, which is in my opinion rather pointless (and maybe unethical).

But I really think that’s not what’s happening. Maybe I’m a good teacher, but at the end of the day I am just a guide. If my students do well, or even if they don’t do well, it is because they spent many hours on the challenges that I designed, and have learned a lot from the whole experience. The credit for any success thus lies solely through the student’s effort. And that experience, I think, is certainly not zero-sum.

# 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

- 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.
- 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.
- 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.

# An apology for HMMT 2016

Median Putnam contestants, willing to devote one of the last Saturdays before final exams to a math test, are likely to receive an advanced degree in the sciences. It is counterproductive on many levels to leave them feeling like total idiots.

— Bruce Reznick, “Some Thoughts on Writing for the Putnam”

Last February I made a big public apology for having caused one of the biggest scoring errors in HMMT history, causing a lot of changes to the list of top individual students. Pleasantly, I got some nice emails from coaches who reminded me that most students and teams do not place highly in the tournament, and at the end of the day the most important thing is that the contestants enjoyed the tournament.

So now I decided I have to apologize for 2016, too.

The story this time is that I inadvertently sent over 100 students home having solved two or fewer problems total, out of 30 individual problems. That year, I was the problem czar for HMMT February 2016, and like many HMMT problem czars before me, had vastly underestimated the difficulty of my own problems.

I think stories like this are a lot worse than people realize; contests are supposed to be a learning experience for the students, and if a teenager shows up to Massachusetts and spends an entire Saturday feeling hopeless for the entire contest, then the flight back to California is going to feel very long. Now imagine having 100 students go through this every single February.

So today I’d like to say a bit about things I’ve picked up since then that have helped me avoid making similar mistakes. I actually think people generally realize that HMMT is too hard, but are wrong about how this should be fixed. In particular, **I think the common approach (and the one I took) of “make problem 1 so easy that almost nobody gets a zero” is wrong**, and I’ll explain here what I think should be done instead.

## 1. Gettable, not gimme

I think just “easy” is the wrong way to think about the beginning problems. At ARML, the problem authors use a finer distinction which I really like:

- A problem is
**gettable**if nearly every contestant feels like they*could have*gotten the problem on a good day. (In particular, problems that require knowledege that not all contestants have are not gettable, even if they are easy with it.) - A problem is a
**gimme**if nearly every contestant actually solves the problem on the contest.

The consensus is always that **the early problems should be gettable but not gimme’s**. You could start every contest by asking the contestant to compute the expected value of 7, but the contestants are going to notice, and it isn’t going to help anyone.

(I guess I should make the point that in order for a problem to be a “gimme”, it would have to be so easy to be almost insulting, because high accuracy on a given problem is really only possible if the level of the problem is significantly below the level of the student. So a gimme would have to be a problem that is way easier than the level of the weakest contestant — you can see why these would be bad.)

In contrast, with a gettable problem, even though some of the contestants will miss it, they’ll often miss it for a reason like 2+3=6. This is a bit unfortunate, but it is still a lot better if the contestant goes home thinking “I made a small arithmetic error, so I have to be more careful” than “there’s no way I could have gotten this, it was hopeless”.

But that brings to me to the next point:

## 2. At the IMO 33% of the problems are gettable

At the IMO, there are two easy problems (one each day), but there are only six problems. So a full one-third of the problems are gettable: we hope that most students attending the IMO can solve either IMO1 or IMO4, even though many will not solve both.

If you are writing HMMT or some similar contest, I think this means **you should think about the opening in terms of the fraction 1/3**, rather than problem 1. For example, at HMMT, I think the czars should strive instead to make the first three or four out of ten problems on each individual test *gettable*: they should be problems every contestant *could* solve, even though some of them will still miss it anyways. Under the pressure of contest, students are going to make all sorts of mistakes, and so it’s important that there are multiple gettable problems. This way, **every student has two or three or four real chances to solve a problem**: they’ll still miss a few, but at least they feel like they could do something.

(Every year at HMMT, when we look back at the tests in hindsight, the first reflex many czars have is to look at how many people got 0’s on each test, and hope that it’s not too many. The fact that this figure is even worth looking at is in my opinion a sign that we are doing things wrong: is 1/10 any better than 0/10, if the kid solved question 1 quickly and then spent the rest of the hour staring at the other nine?)

## 3. Watch the clock

The other thing I want to say is to spend some time thinking about the entire test as a whole, rather than about each problem individually.

To drive the point: I’m willing to bet that **an HMMT individual test with 4 easy, 6 medium, and 0 hard problems could actually work**, even at the top end of the scores. Each medium problem in isolation won’t distinguish the strongest students. But put six of them all together, and you get two effects:

- Students will make mistakes on some of the problems, and by central limit theorem you’ll get a curve anyways.
- Time pressure becomes significantly more important, and the strongest students will come out ahead by simply being faster.

Of course, I’ll never be able to persuade the problem czars (myself included) to not include at least one or two of those super-nice hard problems. But the point is that they’re not actually needed in situations like HMMT, when there are so many problems that it’s hard to *not* get a curve of scores.

One suggestion many people won’t take: if you really want to include some difficulty problems that will take a while, decrease the length of the test. If you had 3 easy, 3 medium, and 1 hard problem, I bet that could work too. One hour is really not very much time.

Actually, this has been experimentally verified. On my HMMT 2016 Geometry test, nobody solved any of problems 8-10, so the test was essentially seven problems long. The gradient of scores at the top and center still ended up being okay. The only issue was that a third of the students solved zero problems, because the easy problems were either error-prone, or else were hit-or-miss (either solved quickly or not at all). Thus that’s another thing to watch out for.

# Lessons from math olympiads

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:

- To build a social network for gifted high school students with similar interests.
- 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

- 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.
- That’s a Zuming quote.
- 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.
- 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. - Relative to what many high school students work on, not compared to research or something.
- 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.

# Some Thoughts on Olympiad Material Design

(This is a bit of a follow-up to the solution reading post last month. Spoiler warnings: USAMO 2014/6, USAMO 2012/2, TSTST 2016/4, and hints for ELMO 2013/1, IMO 2016/2.)

I want to say a little about the process which I use to design my olympiad handouts and classes these days (and thus by extension the way I personally think about problems). The short summary is that my teaching style is centered around **showing connections and recurring themes between problems**.

Now let me explain this in more detail.

## 1. Main ideas

Solutions to olympiad problems can look quite different from one another at a surface level, but typically they center around one or two **main ideas**, as I describe in my post on reading solutions. Because details are easy to work out once you have the main idea, as far as learning is concerned you can more or less throw away the details and pay most of your attention to main ideas.

Thus whenever I solve an olympiad problem, I make a deliberate effort to summarize the solution in a few sentences, such that I basically know how to do it from there. I also make a deliberate effort, whenever I write up a solution in my notes, to structure it so that my future self can see all the key ideas at a glance and thus be able to understand the general path of the solution immediately.

The example I’ve previously mentioned is USAMO 2014/6.

**Example 1** **(USAMO 2014, Gabriel Dospinescu)**

Prove that there is a constant with the following property: If are positive integers such that for all , then

If you look at any complete solution to the problem, you will see a lot of technical estimates involving and the like. But the main idea is very simple: “consider an table of primes and note the small primes cannot adequately cover the board, since ”. Once you have this main idea the technical estimates are just the grunt work that you force yourself to do if you’re a contestant (and don’t do if you’re retired like me).

Thus the study of olympiad problems is reduced to the study of main ideas behind these problems.

## 2. Taxonomy

So how do we come up with the main ideas? Of course I won’t be able to answer this question completely, because therein lies most of the difficulty of olympiads.

But I do have some progress in this way. It comes down to seeing how main ideas are similar to each other. I spend a lot of time trying to **classify the main ideas** into categories or themes, based on how similar they feel to one another. If I see one theme pop up over and over, then I can make it into a class.

I think **olympiad taxonomy** is severely underrated, and generally not done correctly. The status quo is that people do bucket sorts based on the particular *technical details* which are present in the problem. This is correlated with the main ideas, but the two do not always coincide.

An example where technical sort works okay is Euclidean geometry. Here is a simple example: harmonic bundles in projective geometry. As I explain in my book, there are a few “basic” configurations involved:

- Midpoints and parallel lines
- The Ceva / Menelaus configuration
- Harmonic quadrilateral / symmedian configuration
- Apollonian circle (right angle and bisectors)

(For a reference, see Lemmas 2, 4, 5 and Exercise 0 here.) Thus from experience, any time I see one of these pictures inside the current diagram, I think to myself that “this problem feels projective”; and if there is a way to do so I try to use harmonic bundles on it.

An example where technical sort fails is the “pigeonhole principle”. A typical problem in such a class looks something like USAMO 2012/2.

**Example 2** **(USAMO 2012, Gregory Galperin)**

A circle is divided into congruent arcs by points. The points are colored in four colors such that some points are colored Red, some points are colored Green, some points are colored Blue, and the remaining points are colored Yellow. Prove that one can choose three points of each color in such a way that the four triangles formed by the chosen points of the same color are congruent.

It’s true that the official solution uses the words “pigeonhole principle” but that is not really the heart of the matter; the key idea is that you consider all possible rotations and count the number of incidences. (In any case, such calculations are better done using expected value anyways.)

Now why is taxonomy a good thing for learning and teaching? The reason is that building connections and seeing similarities is most easily done by simultaneously presenting several related problems. I’ve actually mentioned this already in a different blog post, but let me give the demonstration again.

Suppose I wrote down the following:

You can tell what each of the ‘s, ‘s, ‘s have in common by looking for a few moments. But what happens if I intertwine them?

This is the same information, but now you have to work much harder to notice the association between the letters and the numbers they’re next to.

This is why, if you are an olympiad student, I strongly encourage you to keep a journal or blog of the problems you’ve done. Solving olympiad problems takes lots of time and so it’s worth it to spend at least a few minutes jotting down the main ideas. And once you have enough of these, you can start to see new connections between problems you haven’t seen before, rather than being confined to thinking about individual problems in isolation. (Additionally, it means you will never have redo problems to which you forgot the solution — learn from my mistake here.)

## 3. Ten buckets of geometry

I want to elaborate more on geometry in general. These days, if I see a solution to a Euclidean geometry problem, then I mentally store the problem and solution into one (or more) buckets. I can even tell you what my buckets are:

- Direct angle chasing
- Power of a point / radical axis
- Homothety, similar triangles, ratios
- Recognizing some standard configuration (see Yufei for a list)
- Doing some length calculations
- Complex numbers
- Barycentric coordinates
- Inversion
- Harmonic bundles or pole/polar and homography
- Spiral similarity, Miquel points

which my dedicated fans probably recognize as the ten chapters of my textbook. (Problems may also fall in more than one bucket if for example they are difficult and require multiple key ideas, or if there are multiple solutions.)

Now whenever I see a new geometry problem, the diagram will often “feel” similar to problems in a certain bucket. Exactly what I mean by “feel” is hard to formalize — it’s a certain gut feeling that you pick up by doing enough examples. There are some things you can say, such as “problems which feature a central circle and feet of altitudes tend to fall in bucket 6”, or “problems which only involve incidence always fall in bucket 9”. But it seems hard to come up with an exhaustive list of hard rules that will do better than human intuition.

## 4. How do problems feel?

But as I said in my post on reading solutions, there are deeper lessons to teach than just technical details.

For examples of themes on opposite ends of the spectrum, let’s move on to combinatorics. Geometry is quite structured and so the themes in the main ideas tend to translate to specific theorems used in the solution. Combinatorics is much less structured and many of the themes I use in combinatorics cannot really be formalized. (Consequently, since everyone else seems to mostly teach technical themes, several of the combinatorics themes I teach are idiosyncratic, and to my knowledge are not taught by anyone else.)

For example, one of the unusual themes I teach is called **Global**. It’s about the idea that to solve a problem, you can just kind of “add up everything at once”, for example using linearity of expectation, or by double-counting, or whatever. In particular these kinds of approach ignore the “local” details of the problem. It’s hard to make this precise, so I’ll just give two recent examples.

**Example 3** **(ELMO 2013, Ray Li)**

Let be nine real numbers, not necessarily distinct, with average . Let denote the number of triples for which . What is the minimum possible value of ?

**Example 4** **(IMO 2016)**

Find all integers for which each cell of table can be filled with one of the letters , and in such a way that:

- In each row and column, one third of the entries are , one third are and one third are ; and
- in any diagonal, if the number of entries on the diagonal is a multiple of three, then one third of the entries are , one third are and one third are .

If you look at the solutions to these problems, they have the same “feeling” of adding everything up, even though the specific techniques are somewhat different (double-counting for the former, diagonals modulo for the latter). Nonetheless, my experience with problems similar to the former was immensely helpful for the latter, and it’s why I was able to solve the IMO problem.

## 5. Gaps

This perspective also explains why I’m relatively bad at functional equations. There are *some* things I can say that may be useful (see my handouts), but much of the time these are just technical tricks. (When sorting functional equations in my head, I have a bucket called “standard fare” meaning that you “just do work”; as far I can tell this bucket is pretty useless.) I always feel stupid teaching functional equations, because I never have many good insights to say.

Part of the reason is that functional equations often don’t have a main idea at all. Consequently it’s hard for me to do useful taxonomy on them.

Then sometimes you run into something like the windmill problem, the solution of which is fairly “novel”, not being similar to problems that come up in training. I have yet to figure out a good way to train students to be able to solve windmill-like problems.

## 6. Surprise

I’ll close by mentioning one common way I come up with a theme.

Sometimes I will run across an olympiad problem which I solve quickly, and think should be very easy, and yet once I start grading I find that the scores are much lower than I expected. Since the way I solve problems is by drawing experience from similar previous problems, this must mean that I’ve subconsciously found a general framework to solve problems like , which is not obvious to my students yet. So if I can put my finger on what that framework is, then I have something new to say.

The most recent example I can think of when this happened was TSTST 2016/4 which was given last June (and was also a very elegant problem, at least in my opinion).

**Example 5** **(TSTST 2016, Linus Hamilton)**

Let be a positive integers. Prove that we must apply the Euler function at least times before reaching .

I solved this problem very quickly when we were drafting the TSTST exam, figuring out the solution while walking to dinner. So I was quite surprised when I looked at the scores for the problem and found out that empirically it was not that easy.

After I thought about this, I have a new tentative idea. You see, when doing this problem I really was thinking about “what does this operation do?”. You can think of as an infinite tuple

of prime exponents. Then the can be thought of as an operation which takes each nonzero component, decreases it by one, and then adds some particular vector back. For example, if then is decreased by one and each of and are increased by one. In any case, if you look at this behavior for long enough you will see that the coordinate is a natural way to “track time” in successive operations; once you figure this out, getting the bound of is quite natural. (Details left as exercise to reader.)

Now when I read through the solutions, I found that many of them had not really tried to think of the problem in such a “structured” way, and had tried to directly solve it by for example trying to prove (which is false) or something similar to this. I realized that had the students just ignored the task “prove ” and spent some time getting a better understanding of the structure, they would have had a much better chance at solving the problem. Why had I known that structural thinking would be helpful? I couldn’t quite explain it, but it had something to do with the fact that the “main object” of the question was “set in stone”; there was no “degrees of freedom” in it, and it was concrete enough that I felt like I could understand it. Once I understood how multiple operations behaved, the bit about almost served as an “answer extraction” mechanism.

These thoughts led to the recent development of a class which I named **Rigid**, which is all about problems where the point is not to immediately try to prove what the question asks for, but to first step back and understand completely how a particular rigid structure (like the in this problem) behaves, and to then solve the problem using this understanding.

# On Reading Solutions

(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” () to “essentially solved” (). 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 , , , be real numbers such that and all zeros , , , and of the polynomial are real. Find the smallest value the product

can take.

*Proof:* (Not-so-good write-up) Since for every (where ), we get which equals to . If this is and . Also, , this is .

*Proof:* (Better write-up) The answer is . This can be achieved by taking , whence the product is , and .

Now, we prove this is a lower bound. Let . The key observation is that

Consequently, we have

This proves the lower bound.

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

which allows you use the enigmatic condition .

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 , achieved by — 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

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 , just based on the condition . 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 , might look as follows.

*Proof:* First, we prove lemma . (Proof of , which is USAMO 1/4 level.)

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

Finally, we remark that putting together and solves the problem.

Likely the main difficulty of is actually *finding* and . So a very experienced student might think of the sub-proofs as “easy details”. But younger students might find 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 were the main ideas to begin with. In that case, the problem 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 with midpoint , and a line parallel to , so I should consider projecting through a point on .
- 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 numbers on a circle, odd. The counterexamples for 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 is a random variable that takes on only nonnegative integer values, with , , and . Determine the smallest possible value of the probability of the event .

**Problem 3** **(Putnam 2014 B2)**

Suppose that is a function on the interval such that for all and

How large can 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 / 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 .

That conjectured 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 . 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 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.

This is a bad idea.

## 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 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 probability of getting each question right. Then the chance he sweeps the AIME is

This is pretty low, which makes sense: and on the AIME feel like quite different scores.

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

This should feel kind of low to you as well. So if you consistently solve 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 problems (thus, success), id est a *near-perfect* score. Then the probability of getting a perfect score

Which is\dots just over .

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

for small . That is, if your accuracy is even a little away from perfect, that difference gets amplified by a factor of 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.

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 , the probability of getting looks to be about . As for 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:

take three or four minutes and do the following quiz.

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 to average 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.

# Stop Paying Me Per Hour

Occasionally I am approached by parents who ask me if I am available to teach their child in olympiad math. This is flattering enough that I’ve even said yes a few times, but I’m always confused why the question is “can you tutor my child?” instead of “do you think tutoring would help, and if so, can you tutor my child?”.

Here are my thoughts on the latter question.

## Charging by Salt

I’m going to start by clearing up the big misconception which inspired the title of this post.

The way tutoring works is very roughly like the following: I meet with the student once every week, with custom-made materials. Then I give them some practice problems to work on (“homework”), which I also grade. I throw in some mock olympiads. I strongly encourage my students to email me with questions as they come up. Rinse and repeat.

The actual logistics vary; for example, for small in-person groups I prefer to do every other week for 3 hours. But the thing that never changes is how the parents pay me. It’s always the same: I get dollars per hour for the actual in-person meeting, and dollars per hour for preparing materials, grading homework, responding to questions, and writing the mock olympiads.

Now I’m not complaining because is embarrassingly large. But one day I realized that this pricing system is giving parents the *wrong impression*. They now think the bulk of the work is from me taking the time to meet with their child, and that the homework is to help reinforce what I talk about in class. After all, this is what high school does, right?

I’m here to tell you that this is completely wrong.

**It’s the other way around: the class is meant to supplement the homework.** Think salt: for most dishes you can’t get away with having zero salt, but you still don’t price a dish based on how much salt is in it. Similarly, you can’t excise the in-person meeting altogether, but the dirty secret is that the classtime isn’t the core component.

I mean, here’s the thing.

- When you take the IMO, you are alone with a sheet of paper that says “Problem 1”, “Problem 2”, “Problem 3”.
- When you do my homework, you are alone with a sheet of paper that says “Problem 1”, “Problem 2”, “Problem 3”.
- When you’re in my class, you get to see my beautiful smiling face plus a sheet of paper that says “Theorem 1”, “Example 2”, “Example 3”.

Which of these is not like the other?

## Active Ingredients

So we’ve established that the main active ingredient is actually you working on problems alone in your room. If so, why do you need a teacher at all?

The answer depends on what the word “need” means. No USA IMO contestant in my recent memory has had a coach, so you don’t *need* a coach. But there are some good reasons why one might be *helpful*.

Some obvious reasons are social:

- Forces you to work regularly; though most top students don’t really have a problem with self-motivation
- You have a person to talk to. This can be nice if you are relatively isolated from the rest of the math community (e.g. due to geography).
- You have someone who will answer your questions. (I can’t tell you how jealous I am right now.)
- Feedback on solutions to problems. This includes student’s written solutions (stylistic remarks, or things like “this lemma you proved in your solution is actually just a special case of X” and so on) as well as explaining solutions to problems the student fails to solve.

In short, it’s **much more engaging to study math with a real person**.

Those reasons don’t depend so much on the instructor’s actual ability. Here are some reasons which do:

- Guidance. An instructor can tell you what things to learn or work on based on their own experience in the past, and can often point you to things that you didn’t know existed.
- It’s a big plus if the instructor has a good taste in problems. Some problems are bad and don’t teach you anything; some (old) problems don’t resemble the flavor of problems that actually appear on olympiads. On the flip side, some problems are very instructive or very pretty, and it’s great if your teacher knows what these are.
- Ideally, also a good taste in topics. For example, I strongly object to classes titled “collinearity and concurrence” because this may as well be called “geometry”, and I think that such global classes are too broad to do anything useful. Conversely, examples of topics I think should be classes but aren’t: “looking at equality cases”, “explicit constructions”, “Hall’s marriage theorem”, “greedy algorithms”. I make this point a lot more explicitly in Section 2 of this blog post of mine.

In short, **you’re also paying for the material and expertise**. Past IMO medalists know how the contest scene works. Parents and (beginning) students less so.

Lastly, the reason which I personally think is most important:

- Conveys strong intuition/heuristics, both globally and for specific problems. It’s hard to give concrete examples of this, but a few global ones I know were particularly helpful for me: “look at maximal things” (Po-Shen Loh on greedy algorithms), “DURR WE WANT STUFF TO CANCEL” (David Yang on FE’s), “use obvious inequalities” (Gabriel Dospinescu on analytic NT), which are take-aways that have gotten me a lot of points. This is also my biggest criteria for evaluating my own written exposition.

You guys know this feeling, I’m sure: when your English teacher assigned you an passage to read, **the fastest way to understand it is to not read the passage but to ask the person sitting next to you what it’s saying.** I think this is in part because most people are awful at writing and don’t even know how to write for other human beings.

The situation in olympiads is the same. I estimate listening to me explain a solution is maybe 4 to 10 times faster than reading the official solution. Turns out that writing up official solutions for contests is a huge chore, so most people just throw a sequence of steps at the reader without even bothering to identify the main ideas. (As a contest organizer, I’m certainly guilty of this laziness too!)

Aside: I think this is a large part of why my olympiad handouts and other writings have been so well-received. Disclaimer: this was supposed to be a list of what makes a good instructor, but due to narcissism it ended up being a list of things I focus on when teaching.

## Caveat Emptor

And now I explain why the top IMO candidates still got by without teachers.

It turns out that the amount of math preparation time that students put in doesn’t seem to be a normal distribution. It’s a **log normal** distribution. And the reason is this: it’s hard to do a really good job on anything you don’t think about in the shower.

Officially, when I was a contestant I spent maybe 20 hours a week doing math contest preparation. But the actual amount of time is higher. The reason is that I would think about math contests more like 24/7. During English class, I would often be daydreaming about the inequality I worked on last night. On the car ride home, I would idly think about what I was going to teach my middle school students the next week. To say nothing of showers: during *my* showers I would draw geometry diagrams on the wall with water on my finger.

So spiritually, I maybe spent 10 times as much time on math olympiads compared to an average USA(J)MO qualifier.

And that factor of 10 is *enormous*. Even if I as a coach can cause you to learn two or three or four times more efficiently, you will still lose to that factor of 10. I’d guess my actual multiplier is somewhere between 2 and 3, so there you go. (*Edit*: this used to say 3 to 4, I think that’s too high now.)

The best I can do is hope that, in addition to making my student’s training more efficient, I also cause my students to like math more.