# 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.

# 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.

# 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 ${c>0}$ with the following property: If ${a, b, n}$ are positive integers such that ${\gcd(a+i, b+j)>1}$ for all ${i, j \in \{0, 1, \dots, n\}}$, then

$\displaystyle \min\{a, b\}> (cn)^n.$

If you look at any complete solution to the problem, you will see a lot of technical estimates involving ${\zeta(2)}$ and the like. But the main idea is very simple: “consider an ${N \times N}$ table of primes and note the small primes cannot adequately cover the board, since ${\sum p^{-2} < \frac{1}{2}}$”. 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 ${432}$ points. The points are colored in four colors such that some ${108}$ points are colored Red, some ${108}$ points are colored Green, some ${108}$ points are colored Blue, and the remaining ${108}$ 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:

$\displaystyle \begin{array}{lll} A1 & B11 & C8 \\ A9 & B44 & C27 \\ A49 & B33 & C343 \\ A16 & B99 & C1 \\ A25 & B22 & C125 \end{array}$

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

$\displaystyle \begin{array}{lllll} B11 & C27 & C343 & A1 & A9 \\ C125 & B33 & A49 & B44 & A25 \\ A16 & B99 & B22 & C8 & C1 \end{array}$

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:

1. Direct angle chasing
2. Power of a point / radical axis
3. Homothety, similar triangles, ratios
4. Recognizing some standard configuration (see Yufei for a list)
5. Doing some length calculations
6. Complex numbers
7. Barycentric coordinates
8. Inversion
9. Harmonic bundles or pole/polar and homography
10. 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 ${a_1,a_2,\dots,a_9}$ be nine real numbers, not necessarily distinct, with average ${m}$. Let ${A}$ denote the number of triples ${1 \le i < j < k \le 9}$ for which ${a_i + a_j + a_k \ge 3m}$. What is the minimum possible value of ${A}$?

Example 4 (IMO 2016)

Find all integers ${n}$ for which each cell of ${n \times n}$ table can be filled with one of the letters ${I}$, ${M}$ and ${O}$ in such a way that:

• In each row and column, one third of the entries are ${I}$, one third are ${M}$ and one third are ${O}$; and
• in any diagonal, if the number of entries on the diagonal is a multiple of three, then one third of the entries are ${I}$, one third are ${M}$ and one third are ${O}$.

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 ${3}$ 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 ${P}$ which I solve quickly, and think should be very easy, and yet once I start grading ${P}$ 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 ${P}$, 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 ${n > 1}$ be a positive integers. Prove that we must apply the Euler ${\varphi}$ function at least ${\log_3 n}$ times before reaching ${1}$.

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 ${\varphi}$ operation do?”. You can think of ${n}$ as an infinite tuple

$\displaystyle \left(\nu_2(n), \nu_3(n), \nu_5(n), \nu_7(n), \dots \right)$

of prime exponents. Then the ${\varphi}$ 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 ${\nu_7(n) > 0}$ then ${\nu_7}$ is decreased by one and each of ${\nu_2(n)}$ and ${\nu_3(n)}$ are increased by one. In any case, if you look at this behavior for long enough you will see that the ${\nu_2}$ coordinate is a natural way to “track time” in successive ${\varphi}$ operations; once you figure this out, getting the bound of ${\log_3 n}$ 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 ${\varphi(n) \ge n/3}$ (which is false) or something similar to this. I realized that had the students just ignored the task “prove ${n \le 3^k}$” and spent some time getting a better understanding of the ${\varphi}$ 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 ${\varphi}$ operations behaved, the bit about ${\log_3 n}$ 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 ${\varphi}$ in this problem) behaves, and to then solve the problem using this understanding.

# 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 $N \gg 0$ dollars per hour for the actual in-person meeting, and $0$ dollars per hour for preparing materials, grading homework, responding to questions, and writing the mock olympiads.

Now I’m not complaining because $N$ 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.

# On Problem Sets

(It appears to be May 7 — good luck to all the national MathCounts competitors tomorrow!)

1. An 8.044 Problem

Recently I saw a 8.044 physics problem set which contained the problem

Consider a system of ${N}$ almost independent harmonic oscillators whose energy in a microcanonical ensemble is given by ${E = \frac 12 \hbar \omega N + \hbar \omega M}$. Show that this energy can be obtained is ${\frac{(M+N-1)!}{M!(N-1)!}}$.

Once you remove the physics fluff, it immediately reduces to

Show the number of nonnegative integer solutions to ${M = \sum_{i=1}^N n_i}$ is ${\frac{(M+N-1)!}{M!(N-1)!}}$.

And as anyone who has done lots of math contests knows, this is the famous stars and bars problem (also known as balls and urns).

This made me really upset when I saw it, for two reasons. One, the main difficulty of the question isn’t related to the physics at hand at all. Once you plug in the definition you get a fairly elegant combinatorics problem, not a physics problem. And secondly, although the solution to the (unrelated) combinatorics is nice, it’s very tricky. I don’t think I could have come up with it easily if I hadn’t seen it before. Either you’ve seen the stars-and-bars trick before and the problem is trivial, or you haven’t seen the trick, and you could easily spend a couple hours trying to come up with a solution — and none of that two hours is teaching you any physics.

You can see why a physics instructor might give this as a homework problem. The solution is short and elementary, something that a undergraduate student could understand and write down. But somewhere at MIT, some poor non-mathematician just spent a good chunk of their evening struggling with this one-trick classic and probably not learning much from it.

2. Don’t I Like Hard Problems?

Well, “not learning much from it” is not entirely accurate\dots

Something that bothered me (and which I hope also bothers the reader) was I complained that the problem was “tricky”. That seems off, because as you might already know, I like hard problems; in fact, in high school I was well despised for helping teachers find hard extra credit problems to pose. (“Hard” isn’t quite the same as “tricky”, but that’s a different direction altogether.) After all, hard problems from math contests taught me to think, isn’t that right?

Well, maybe what’s wrong is that there’s no physics in the hard part of the problem; the bonus problems I provided for my teachers were all closely tied to the material at hand. But that doesn’t seem right either. Euclidean geometry might be useless outside of high school, but nonetheless all the time I spent developing barycentric coordinates still made me a smarter person. Similarly, Richard Rusczyk will often tell you that geometry problems trained him for running the business that is now the Art of Problem Solving. For exactly the same reason, thinking about the stars and bars problem is certainly good for the mind, isn’t that right? Why was I upset about it?

Well, I still hold my objection that there’s no physics in the problem. Why? So at this point we’re naturally led to ask: what was the point of the problem set in the first place? And that answer this, you have to ask: what was the point of the class in the first place?

On paper, it’s to learn physics. Is that really all? Maybe the professor thinks it’s important to teach students how to think as well. Does she? And the answer here is I really don’t know, because I have no idea who’s teaching the class. So I’ll instead ask the more idealistic question: should she?

And surprisingly, I think the answer can be very different from place to place.

On one extreme, I think high school math should be mainly about teaching students to think. Virtually none of the students will actually use the specific content being taught in the class. Why does the average high school student need to know what ${\int_{[0,1]} x^2 \; dx}$ is? They don’t, and that shouldn’t be the point of the class; not the least of reasons being that in ten years half of them won’t even remember what ${\int}$ means anymore.

But on the other extreme, if you have a math major trying to learn the undergraduate curriculum the picture can change entirely, just because there is so much math to cover. It’s kind of ridiculous, honestly: take the average incoming freshman and the average senior math major, and the latter will know so much more than the former. So in this case I would be much more worried about the content of the course; assuming for example that I’m hoping to be a math major, the chance that the (main ideas of) the specific content will be useful later on is far higher.

This is especially true for, say, students who did math contests extensively in high school, because that ability to solve hard problems is already there; it’s not an interesting use of time to be slowly doing challenging exercises in group theory when there’s still modules, rings, fields, categories, algebraic geometry, homological algebra, all untouched (to say nothing of analysis).

What this boils down to is trying to distinguish between the actual content of the given class (something very local) versus the more general skill of problem-solving or thinking. In high school I focused almost exclusively on the latter; as time passes I’ve been shifting my focus farther and farther to the former.

3. ${\text{A} \ge 90\%}$

Now suppose that we are interested in teaching how to think on these problem sets. There’s one other difference between the problem sets and math contests. You’re expected to finish your problem sets and you’re not expected to finish math contests.

I want to complain that there seems to be a stigma that you have to do exercises in order to learn math or physics or whatever, and that people who give up on them are somehow lazy or something. It is true in some sense that you can only learn math by doing. It is probably true that thinking about a hard problem will teach you something. What is not true is that you should always stare at a problem until either it or you cracks.

This is obviously true in math contests too. One of the things I was really bad at was giving up on a problem after hours of no progress. In some sense the time limit of contests is kind of nice; it cuts you off from spending too long on any one problem. You can’t be expected to be able to solve all hard problems, or else they’re not hard.

Problem sets fare much more poorly in this respect. The benefit of thinking about the hard problem diminishes over time (e.g. a typical exercise can teach you more in the first hour than it does in the next six) and sometimes you’re just totally dead in the water after a couple hours of staring. The big guy seem to implicitly tell you that you should keep working because it’s supposed to be hard. Is that really true? It certainly wasn’t true in the math contest world, so I don’t see any reason why it’s true here.

In other words, I don’t think our poor physics student would have lost much by giving up on balls and urns after a few hours. And really, for all the warnings that looking up problems online is immoral, is asking your friend to help really that different?

# Teaching A* USAMO Camp

In the last week of December I got a position as the morning instructor for the A* USAMO winter camp. Having long lost interest in coaching for short-answer contests, I’d been looking forward to an opportunity to teach an olympiad class for ages, and so I was absolutely psyched for that week. In this post I’ll talk about some of the thoughts I had while teaching, in no particular order.

1. Class Format

Here were the constraints I was working with. After removing guest lectures, exams, and so on I had four days of teaching time, one for each of the four olympiad subjects (algebra, geometry, combinatorics, number theory). I taught the morning session, meaning I had a three-hour block each day (with a 15-minute break). I had a wonderfully small class — just five students.

Here’s the format I used for the class, which seemed to work reasonably well (as in, if I were to teach the class again I would probably not change it very much.)

• (0:00-0:10) I usually started the class with a quick warm-up problem (something pretty easy), just to soak up time from latecomers and give students a chance to get ready and glance through the handout. (If you give smart students a pretty handout, the first thing they will do is look through it, regardless of what you tell them to actually do.)
• (0:10-1:30) Afterwards I would go through the lecture, both theory and examples, up until the break. On average this got split up with about half the time for the theory and half the time for the examples. I typically let students try the examples themselves for five minutes (again, smart students will automatically start on the problems regardless of whether you tell them to or not) before I discuss the solution, just so they at least have a feeling for what it is — I consider it immoral to start talking about a solution before students have had a chance to try a problem.
• (1:30-2:40) After a break, I would give the students a long period (a little over an hour) to try the practice problems in the last section of the handout. Since the class was so small, I would prepare about 5-7 practice problems and then let each of them pick a different problem to start working on. (Once they solved their own problem, they would go on and try other ones.)During this time, I was able to take advantage of the small class size in a pretty great way: throughout the hour I would walk around the room talking to each of the students about the problem they were working on. In particular, I tried to make sure every student at least solved the problem they started with.
• (2:40-3:00) In the last 20 minutes of class or so, I had each student present the solution to the problem which they worked on. I think the main utility of this is that it forces the present-er to know clearly in their head what their solution is. This was actually possibly more useful for my feedback than for the students: if a student could present the solution to their problem to the class then I knew they understood at least each of the individual steps.Overall I think this format did more or less what I intended it to do, and will definitely be re-using it if I ever teach an olympiad class in this style again. Though I don’t know how well the second half might work in a bigger room: I actually had to do a bit of running to keep up with questions and ideas that the students came up with while working, and of course the presentation time is proportional to the number of students you have (maybe 3-5 minutes each). So if I had, say, 10 students, I would probably re-think how to run the end.

2. Picking Topics

I think it’s general kind of useless to teach a class where you do a mix of unrelated problems. For example, I never really liked “functional equations” as a class. And don’t even get me started on the typical “divisibility” class. That’s what the IMO Shortlist is for, and the students already have that. Anyone competing at this level already knows how to pick up a collection of problems and practice against it. Class needs to something more than that.

My idea is that problems in an olympiad class should be linked by some underlying, specific theme. It doesn’t have to be a specific technique, but it can be. The reason is that this way, you can see the theme re-appear over and over again. By the time you see it the fifth time, hopefully things start to click.

Let me phrase this another way. Suppose I gave wrote down the following:

$\displaystyle \begin{array}{lll} A1 & B11 & C8 \\ A9 & B44 & C27 \\ A49 & B33 & C343 \\ A16 & B99 & C1 \\ A25 & B22 & C125 \end{array}$

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

$\displaystyle \begin{array}{lllll} B11 & C27 & C343 & A1 & A9 \\ C125 & B33 & A49 & B44 & A25 \\ A16 & B99 & B22 & C8 & C1 \end{array}$

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.

I think the class is kind of the same idea. If you want to draw out the idea of orders, pick a bunch of problems that involve orders in spirit. They don’t have to be exactly the same problem, but they should be reasonably related.

So to produce a good olympiad handout, you need to have something to say. I think my Chinese Remainder Theorem handout is a good example (it was actually something I was considering using for the NT session, but I decided on something else eventually). I want to point out how CRT is used in constructions, so the examples and practice problems are all designed to illustrate this point. There’s a large degree of micro-control throughout the entire thing.

Honestly, I think it’s really easy to teach olympiad math badly: just pick a bunch of unrelated problems, go through the solutions one by one, then give some more unrelated problems for practice. The students will still get better, because they are practicing. But is that all you can do as a teacher?

For the record, here’s the topics I ended up using for the camp.

• Orders / Lifting the Exponent
• Irreducibility of Polynomials
• Projective Geometry
• Double Summation

3. Narrowing Problems

Something new I tried for this lecture was trimming a lot.

At MOP, I’d often get a handout for a MOP class with something like 30 problems on it. We’d get to pick which ones we worked on, and then we’d see or present some of the solutions in class. The issue is that, well, a class isn’t that long, so I would only be able to work on two or three problems, and these wouldn’t be the same as the two or three problems other people worked on or presented.

I think the hope was that when we went home we’d still have like 20 various problems to work on. The problem is that I couldn’t possibly have worked on the left-over 20 problems from every class even if I wanted to — there were just too many.

I fought this issue at A* by trimming down the practice problems a lot. My handouts essentially had only 5-7 problems to work on. This way, more people had looked at the same problems when it came time for presentations.

The reason I picked 5-7 was so that every student could work on (and hence present) a different problem. In retrospect I’m not sure this was a good idea. If I were to teach again, I might even cut it down to fewer than that, maybe four problems or so. That way, everyone really works on the same problems, and presentations of solutions are infinitely more useful. I would just have to work around the fact that on any given day, not all students would have a chance to present.

Finally, here’s a couple things I wish I had fixed.

First, I made a lot of assumptions about what people knew and didn’t knew. I thought I had made the NT lecture too hard because the room was very quiet, but it in fact turned out that it was because the students had actually seen most of the order material before.

The only reason I found out was because after I had finished presenting all the order material, I asked out of curiosity whether anyone had seen this already, not actually expecting anyone to raise their hands. Instead, the entire class did. Students really are too polite — I must have bored them to tears for those first 45 minutes.

The solution to this is really simple: just ask the students if they’ve seen it before. Any teacher knows that students are shy to admitting they don’t understand what you’re talking about, but if you just ask “have any of you seen this before?” the students will in general be pretty honest. (If you phrase it as “have any of you not seen this before” the results are less accurate.) So that’s something I will remember to do much more of later on.

The other thing is that I likely made the practice problems too hard. I felt like I had to give too many hints: at least the students understand the solution, but I’m not sure how helpful it is to only solve the problem because the hints given amounted to an outline of the solution. In my defense, I was guessing in the dark as to the abilities of the students, and erred on the side of hard. (Any of you who do math olympiads know how useless and boring a too-easy class is; in contrast, classes which are a tad too hard can often still be beneficial.) But the point stands that my estimate was wrong.

Finally, I think I wore the students out a bit too much. I was happy with their performance in my class, but apparently they were all pretty tired during afternoon. But I think that might just be because of the way the camp is set up — six hours of class a day is really a lot, even for the very hardcore.

5. Closings

Overall I was quite happy with how the classes turned out, and I think the students were too (either that or they were very generous with my instructor ratings). I can’t wait until I get an opportunity like this again, but that might be a long time in coming — there really aren’t that many USAMO-level students out there as I would like!