I used to 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.)

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*? What kind of message does that send to our young contestants?

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.

]]>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

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.

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]

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.

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.

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”.

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**.

- 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.

One of the major headaches of using complex numbers in olympiad geometry problems is dealing with square roots. In particular, it is nontrivial to express the incenter of a triangle inscribed in the unit circle in terms of its vertices.

The following lemma is the standard way to set up the arc midpoints of a triangle. It appears for example as part (a) of Lemma 6.23.

**Theorem 1** **(Arc midpoint setup for a triangle)**

Let be a triangle with circumcircle and let , , denote the arc midpoints of opposite , opposite , opposite .

Suppose we view as the unit circle in the complex plane. Then *there exist* complex numbers , , such that , , , and

Theorem 1 is often used in combination with the following lemma, which lets one assign the incenter the coordinates in the above notation.

**Lemma 2** **(The incenter is the orthocenter of opposite arc midpoints)**

Let be a triangle with circumcircle and let , , denote the arc midpoints of opposite , opposite , opposite . Then the incenter of coincides with the orthocenter of .

Unfortunately, the proof of Theorem 1 in my textbook is wrong, and I cannot find a proof online (though I hear that *Lemmas in Olympiad Geometry* has a proof). So in this post I will give a correct proof of Theorem 1, which will hopefully also explain the mysterious introduction of the minus signs in the theorem statement. In addition I will give a version of the theorem valid for quadrilaterals.

I should at once warn the reader that Theorem 1 is an *existence result*, and thus must be applied carefully.

To see why this matters, consider the following problem, which appeared as problem 1 of the 2016 JMO.

**Example 3** **(JMO 2016, by Zuming feng)**

The isosceles triangle , with , is inscribed in the circle . Let be a variable point on the arc that does not contain , and let and denote the incenters of triangles and , respectively. Prove that as varies, the circumcircle of triangle passes through a fixed point.

By experimenting with the diagram, it is not hard to guess that the correct fixed point is the midpoint of arc , as seen in the figure below. One might be tempted to write , , , and assert the two incenters are and , and that the fixed point is .

This is a mistake! If one applies Theorem 1 twice, then the choices of “square roots” of the common vertices and may not be compatible. In fact, they *cannot* be compatible, because the arc midpoint of opposite is different from the arc midpoint of opposite .

In fact, I claim this is not a minor issue that one can work around. This is because the claim that the circumcircle of passes through the midpoint of arc is false if lies on the arc on the same side as ! In that case it actually passes through instead. Thus the truth of the problem really depends on the fact that the quadrilateral is *convex*, and any attempt with complex numbers must take this into account to have a chance of working.

Fix now, so we require , , . There are choices of square roots , , we can take (differing by a sign); we wish to show one of them works.

We pick an arbitrary choice for first. Then, of the two choices of , we pick the one such that . Similarly, for the two choices of , we pick the one such that . Our goal is to show that under these conditions, we have again.

The main trick is to now consider the arc midpoint , which we denote by . It is easy to see that:

**Lemma 4** **(The isosceles trapezoid trick)**

We have (both are perpendicular to the bisector). Thus is an isosceles trapezoid, and so .

Thus, we have

Thus

as desired.

From this we can see why the minus signs are necessary.

We now return to the setting of a convex quadrilateral that we encountered in Example 3. Suppose we preserve the variables , , that we were given from Theorem 1, but now add a fourth complex number with . How are the new arc midpoints determined? The following theorem answers this question.

**Theorem 6** **( setup)**

Let be a convex quadrilateral inscribed in the unit circle of the complex plane. Then we can choose complex numbers , , , such that , , , and:

- The opposite arc midpoints , , of triangle are given by , , , as before.
- The midpoint of arc not including or is given by .
- The midpoint of arc not including or is given by .
- The midpoint of arc is and the midpoint of arc is .

This setup is summarized in the following figure.

Note that unlike Theorem 1, the four arcs cut out by the sides of do not all have the same sign (I chose to have coordinates ). This asymmetry is inevitable (see if you can understand why from the proof below).

*Proof:* We select , , with Theorem 1. Now, pick a choice of such that is the arc midpoint of not containing and . Then the arc midpoint of not containing or is given by

On the other hand, the calculation of for the midpoint of follows by applying Lemma 4 again. (applied to triangle ). The midpoint of is computed similarly.

In other problems, the four vertices of the quadrilateral may play more symmetric roles and in that case it may be desirable to pick a setup in which the four vertices are labeled in order. By relabeling the letters in Theorem 6 one can prove the following alternate formulation.

**Corollary 7**

Let be a convex quadrilateral inscribed in the unit circle of the complex plane. Then we can choose complex numbers , , , such that , , , and:

- The midpoints of , , , cut out by the sides of are , , , .
- The midpoints of and are and .
- The midpoints of and are and .

To test the newfound theorem, here is a cute easy application.

**Example 8** **(Japanese theorem for cyclic quadrilaterals)**

In a cyclic quadrilateral , the incenters of , , , are the vertices of a rectangle.

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.

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.)

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.

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.

- 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.

This work was part of the Duluth REU 2017, and I thank Joe Gallian for suggesting the problem.

Let me begin by formulating the problem as it was given to me. First, here is the definition and notation for a “block-ascending” permutation.

**Definition 1**

For nonnegative integers , …, an *-ascending permutation* is a permutation on whose descent set is contained in . In other words the permutation ascends in blocks of length , , …, , and thus has the form

for which for all .

It turns out that block-ascending permutations which also avoid an increasing subsequence of certain length have nice enumerative properties. To this end, we define the following notation.

**Definition 2**

Let denote the set of -ascending permutations which avoid the pattern .

(The reason for using will be explained later.) In particular, if .

**Example 3**

Here is a picture of a permutation in (but not in , since one can see an increasing length subsequence shaded). We would denote it .

Now on to the results. A 2011 paper by Joel Brewster Lewis (JBL) proved (among other things) the following result:

**Theorem 4** **(Lewis 2011)**

The sets and are in bijection with Young tableau of shape .

**Remark 5**

When , this implies , which is the set of -avoiding permutations of length , is in bijection with the Catalan numbers; so is which is the set of -avoiding *zig-zag* permutations.

Just before the Duluth REU in 2017, Mei and Wang proved that in fact, in Lewis’ result one may freely mix and ‘s. To simplify notation,

**Definition 6**

Let . Then denotes where

**Theorem 7** **(Mei, Wang 2017)**

The sets are also in bijection with Young tableau of shape .

The proof uses the RSK correspondence, but the authors posed at the end of the paper the following open problem:

**Problem
**

Find a direct bijection between the sets above, not involving the RSK correspondence.

This was the first problem that I was asked to work on. (I remember I received the problem on Sunday morning; this actually matters a bit for the narrative later.)

At this point I should pause to mention that this notation is my own invention, and did not exist when I originally started working on the problem. Indeed, all the results are restricted to the case where for each , and so it was unnecessary to think about other possibilities for : Mei and Wang’s paper use the notation . So while I’ll continue to use the notation in the blog post for readability, it will make some of the steps more obvious than they actually were.

Mei and Wang’s paper originally suggested that rather than finding a bijection for any and , it would suffice to biject

and then compose two such bijections. I didn’t see why this should be much easier, but it didn’t seem to hurt either.

As an example, they show how to do this bijection with and . Indeed, suppose . Then is an increasing sequence of length right at the start of . So had better be the largest element in the permutation: otherwise later in the biggest element would complete an ascending permutation of length already! So removing gives a bijection between .

But if you look carefully, this proof does essentially nothing with the later blocks. The exact same proof gives:

**Proposition 8**

Suppose . Then there is a bijection

by deleting the st element of the permutation (which must be largest one).

Once I found this proposition I rejected the initial suggestion of specializing . The “easy case” I had found told me that I could take a set and delete the single element from it. So empirically, my intuition from this toy example told me that it would be easier to find bijections whee and were only “a little different”, and hope that the resulting bijection only changed things a little bit (in the same way that in the toy example, all the bijection did was delete one element). So I shifted to trying to find small changes of this form.

I had a lucky break of wishful thinking here. In the notation with , I had found that one could replace with either or freely. (But this proof relied heavily on the fact the block really being on the far left.) So what other changes might I be able to make?

There were two immediate possibilities that came to my mind.

**Deletion**: We already showed could be changed from to for any . If we can do a similar deletion with for any , not just , then we would be done.**Swapping**: If we can show that two adjacent ‘s could be swapped, that would be sufficient as well. (It’s also possible to swap non-adjacent ‘s, but that would cause more disruption for no extra benefit.)

Now, I had two paths that both seemed plausible to chase after. How was I supposed to know which one to pick? (Of course, it’s possible neither work, but you have to start somewhere.)

Well, maybe the correct thing to do would have to just try both. But it was Sunday afternoon by the time I got to this point. Granted, it was summer already, but I knew that come Monday I would have doctor appointments and other trivial errands to distract me, so I decided I should pick one of them and throw the rest of the day into it. But that meant I had to pick one.

(I confess that I actually already had a prior guess: the deletion approach seemed less likely to work than the swapping approach. In the deletion approach, if is somewhere in the middle of the permutation, it seemed like deleting an element could cause a lot of disruption. But the swapping approach preserved the total number of elements involved, and so seemed more likely that I could preserve structure. But really I was just grasping at straws.)

Yeah, I cheated. Sorry.

Those of you that know anything about my style of math know that I am an algebraist by nature — sort of. It’s more accurate to say that I depend on having concrete examples to function. True, I can’t do complexity theory for my life, but I also haven’t been able to get the hang of algebraic geometry, despite having tried to learn it three or four times by now. But enumerative combinatorics? OH LOOK EXAMPLES.

Here’s the plan: let . Then using a C++ computer program:

- Enumerate all the permutations in .
- Enumerate all the permutations in .
- Enumerate all the permutations in .

If the deletion approach is right, then I would hope and look pretty similar. On the flip side, if the swapping approach is right, then and should look close to each other instead.

It’s moments like this where my style of math really shines. I don’t have to make decisions like the above off gut-feeling: do the “data science” instead.

Except this isn’t actually what I did, since there was one problem. Computing the longest increasing subsequence of a length permutation takes time, and there are or so permutations. But when , we have , which is a pretty big number. Unfortunately, my computer is not really that fast, and I didn’t really have the patience to implement the “correct” algorithms to bring the runtime down.

The solution? Use instead.

In a deep irony that I didn’t realize at the time, it was this moment when I introduced the notation, and for the first time allowed the to not be in . My reasoning was that since I was only doing this for heuristic reasons, I could instead work with and probably not change much about the structure of the problem, while replacing , which would run times faster. This was okay since all I wanted to do was see how much changing the “middle” would disrupt the structure.

And so the new plan was:

- Enumerate all the permutations in .
- Enumerate all the permutations in .
- Enumerate all the permutations in .

I admit I never actually ran the enumeration with , because the route with and turned out to be even more promising than I expected. When I compared the empirical data for the sets and , I found that the number of permutations with any particular triple were equal. In other words, the **outer blocks were preserved**: the bijection

does not tamper with the outside blocks of length and .

This meant I was ready to make the following conjecture. Suppose , . There is a bijection

which only involves rearranging the elements of the th and st blocks.

At this point I was in a quite good position. I had pinned down the problem to a finding a particular bijection that I was confident had to exist, since it was showing up to the empirical detail.

Let’s call this mythical bijection . How could I figure out what it was?

Let me quickly introduce a definition.

**Definition 9**

We say two words and are *order-isomorphic* if if and only . Then order-isomorphism gives equivalence classes, and there is a canonical representative where the letters are ; this is called a *reduced* word.

**Example 10**

The words , and are order-isomorphic; the last is reduced.

Now I guessed one more property of : this should order-isomorphism.

What do I mean by this? Suppose in one context changed to ; then we would expect that in another situation we should have changing to . Indeed, we expect (empirically) to not touch surrounding outside blocks, and so it would be very strange if behaved differently due to far-away numbers it wasn’t even touching.

So actually I’ll just write

for this example, reducing the words in question.

With this hunch it’s possible to cheat with C++ again. Here’s how.

Let’s for concreteness suppose and the particular sets

Well, it turns out if you look at the data:

- The only element of which starts with and ends with is .
- The only element of which starts with and ends with is .

So that means that is changed to . Thus the empirical data shows that

In general, it might not be that clear cut. For example, if we look at the permutations starting with and , there is more than one.

- and are both in .
- and are both in in .

Thus

but we can’t tell which one goes to which (although you might be able to guess).

Fortunately, there is *lots of data*. This example narrowed down to two values, but if you look at other places you might have different data on . Since we think is behaving the same “globally”, we can piece together different pieces of data to get narrower sets. Even better, is a bijection, so once we match either of or , we’ve matched the other.

You know what this sounds like? Perfect matchings.

So here’s the experimental procedure.

- Enumerate all permutations in and .
- Take each possible tuple , and look at the permutations that start and end with those particular four elements. Record the reductions of and for all these permutations. We call these
*input words*and*output words*, respectively. Each output word is a “candidate” of for a input word. - For each input word that appeared, take the intersection of all output words that appeared. This gives a bipartite graph , with input words being matched to their candidates.
- Find perfect matchings of the graph.

And with any luck that would tell us what is.

Luckily, the bipartite graph is quite sparse, and there was only one perfect matching.

246|1357 => 2467|135 247|1356 => 2457|136 256|1347 => 2567|134 257|1346 => 2357|146 267|1345 => 2367|145 346|1257 => 3467|125 347|1256 => 3457|126 356|1247 => 3567|124 357|1246 => 1357|246 367|1245 => 1367|245 456|1237 => 4567|123 457|1236 => 1457|236 467|1235 => 1467|235 567|1234 => 1567|234

If you look at the data, well, there are some clear patterns. Exactly one number is “moving” over from the right half, each time. Also, if is on the right half, then it always moves over.

Anyways, if you stare at this for an hour, you can actually figure out the exact rule:

**Claim 11**

Given an input , move if is the largest index for which , or if no such index exists.

And indeed, once I have this bijection, it takes maybe only another hour of thinking to verify that this bijection works as advertised, thus solving the original problem.

Rather than writing up what I had found, I celebrated that Sunday evening by playing Wesnoth for 2.5 hours.

On Monday morning I was mindlessly feeding inputs to the program I had worked on earlier and finally noticed that in fact and also had the same cardinality. Huh.

It seemed too good to be true, but I played around some more, and sure enough, the cardinality of seemed to only depend on the order of the ‘s. And so at last I stumbled upon the final form the conjecture, realizing that all along the assumption that I had been working with was a red herring, and that the bijection was really true in much vaster generality. There is a bijection

which only involves rearranging the elements of the th and st blocks.

It also meant I had more work to do, and so I was now glad that I hadn’t written up my work from yesterday night.

I re-ran the experiment I had done before, now with . (This was interesting, because the elements in question could now have either longest increasing subsequence of length , or instead of length .)

The data I obtained was:

246|13578 => 24678|135 247|13568 => 24578|136 248|13567 => 24568|137 256|13478 => 25678|134 257|13468 => 23578|146 258|13467 => 23568|147 267|13458 => 23678|145 268|13457 => 23468|157 278|13456 => 23478|156 346|12578 => 34678|125 347|12568 => 34578|126 348|12567 => 34568|127 356|12478 => 35678|124 357|12468 => 13578|246 358|12467 => 13568|247 367|12458 => 13678|245 368|12457 => 13468|257 378|12456 => 13478|256 456|12378 => 45678|123 457|12368 => 14578|236 458|12367 => 14568|237 467|12358 => 14678|235 468|12357 => 12468|357 478|12356 => 12478|356 567|12348 => 15678|234 568|12347 => 12568|347 578|12346 => 12578|346 678|12345 => 12678|345

Okay, so it looks like:

- exactly two numbers are moving each time, and
- the length of the longest run is preserved.

Eventually, I was able to work out the details, but they’re more involved than I want to reproduce here. But the idea is that you can move elements “one at a time”: something like

while preserving the length of increasing subsequences at each step.

So, together with the easy observation from the beginning, this not only resolves the original problem, but also gives an elegant generalization. I had now proved:

**Theorem 12**

For any , …, , the cardinality

does not depend on the order of the ‘s.

Whenever I look back on this, I can’t help thinking just how incredibly lucky I got on this project.

There’s this perpetual debate about whether mathematics is discovered or invented. I think it’s results like this which make the case for “discovered”. I did not really construct the bijection myself: it was “already there” and I found it by examining the data. In another world where did not exist, all the creativity in the world wouldn’t have changed anything.

So anyways, that’s the behind-the-scenes tour of my favorite combinatorics paper.

]]>Suppose you are a math PhD student at MIT. Officially, this “costs” $50K a year in tuition. Fortunately this number is meaningless, because math PhD students serve time as teaching assistants in exchange for having the nominal sticker price waived. MIT then provides a stipend of about $25K a year for these PhD student’s living expenses. This stipend is taxable, but it’s small and you’d pay only $1K-$2K in federal taxes (about 6%).

The new GOP tax proposal strikes 26 U.S. Code 117(d) which would cause the $50K tuition waiver to *also* become taxable income: the PhD student would pay taxes on an “income” of $75K, at tax brackets of 12% and 25%. If I haven’t messed up the calculation, for our single PhD student this means **paying $10K in federal taxes out of the same $25K stipend (about 40%)**.

I think a 40% tax rate for a PhD student is a *bit* unreasonable; the remaining $15K a year is not too far from the poverty line.

(The relevant sentence is page 96, line 20 of the GOP tax bill.)

]]>**Theorem 1** **(Cayley’s Formula)**

The number of trees on labelled vertices is .

*Proof:* We are going to construct a bijection between

- Functions (of which there are ) and
- Trees on with two distinguished nodes and (possibly ).

This will imply the answer.

Let’s look at the first piece of data. We can visualize it as points floating around, each with an arrow going out of it pointing to another point, but possibly with many other arrows coming into it. Such a structure is apparently called a **directed pseudoforest**. Here is an example when .

You’ll notice that in each component, some of the points lie in a cycle and others do not. I’ve colored the former type of points blue, and the corresponding arrows magenta.

Thus a directed pseudoforest can also be specified by

- a choice of some vertices to be in cycles (blue vertices),
- a permutation on the blue vertices (magenta arrows), and
- attachments of trees to the blue vertices (grey vertices and arrows).

Now suppose we take the same information, but replace the *permutation* on the blue vertices with a *total ordering* instead (of course there are an equal number of these). Then we can string the blue vertices together as shown below, where the green arrows denote the selected total ordering (in this case ):

This is exactly the data of a tree on the vertices with two distinguished vertices, the first and last in the chain of green (which could possibly coincide).

]]>(Meta point re blog: I’m probably going to start posting more and more of these more high-level notes/sketches on this blog on topics that I’ve been just learning. Up til now I’ve been mostly only posting things that I understand well and for which I have a very polished exposition. But the perfect is the enemy of the good here; given that I’m taking these notes for my own sake, I may as well share them to help others.)

**Definition 1**

For us a **quadratic form** is a polynomial , where , , are some integers. We say that it is **primitive** if .

For example, we have the famous quadratic form

As readers are probably aware, we can say a lot about exactly which integers can be represented by : by **Fermat’s Christmas theorem**, the primes (and ) can all be written as the sum of two squares, while the primes cannot. For convenience, let us say that:

**Definition 2**

Let be a quadratic form. We say it **represents** the integer if there exists with . Moreover, **properly represents** if one can find such and which are also relatively prime.

The basic question is: **what can we say about which primes/integers are properly represented by a quadratic form?** In fact, we will later restrict our attention to “positive definite” forms (described later).

For example, Fermat’s Christmas theorem now rewrites as:

**Theorem 3** **(Fermat’s Christmas theorem for primes)**

An odd prime is (properly) represented by if and only if .

The proof of this is classical, see for example my olympiad handout. We also have the formulation for odd integers:

**Theorem 4** **(Fermat’s Christmas theorem for odd integers)**

An odd integer is properly represented by if and only if all prime factors of are .

*Proof:* For the “if” direction, we use the fact that is multiplicative in the sense that

For the “only if” part we use the fact that if a multiple of a prime is properly represented by , then so is . This follows by noticing that if (and ) then .

Tangential remark: the two ideas in the proof will grow up in the following way.

- The fact that “multiplies nicely” will grow up to become the so-called
**composition**of quadratic forms. - The second fact will
**not**generalize for an arbitrary form . Instead, we will see that if a multiple of is represented by a form then some form of the same “discriminant” will represent the prime , but this form need not be the same as itself.

The first thing we should do is figure out when two forms are essentially the same: for example, and should clearly be considered the same. More generally, if we think of as acting on and is any automorphism of , then should be considered the same as . Specifically,

**Definition 5**

Two forms and said to be **equivalent** if there exists

such that . We have and so we say the equivalence is

- a
**proper equivalence**if , and - an
**improper equivalence**if .

So we generally will only care about forms up to proper equivalence. (It will be useful to distinguish between proper/improper equivalence later.)

Naturally we seek some invariants under this operation. By far the most important is:

**Definition 6**

The **discriminant** of a quadratic form is defined as

The discriminant is invariant under equivalence (check this). Note also that we also have .

Observe that we have

So if and (thus too) then for all . Such quadratic forms are called **positive definite**, and we will restrict our attention to these forms.

Now that we have this invariant, we may as well classify equivalence classes of quadratic forms for a fixed discriminant. It turns out this can be done explicitly.

**Definition 7**

A quadratic form is **reduced** if

- it is primitive and positive definite,
- , and
- if either or .

**Exercise 8**

Check there only finitely many reduced forms of a fixed discriminant.

Then the big huge theorem is:

**Theorem 9** **(Reduced forms give a set of representatives)**

Every primitive positive definite form of discriminant is properly equivalent to a unique reduced form. We call this the **reduction** of .

*Proof:* Omitted due to length, but completely elementary. It is a reduction argument with some number of cases.

Thus, for any discriminant we can consider the set

which will be the equivalence classes of positive definite of discriminant . By abuse of notation we will also consider it as the set of equivalence classes of primitive positive definite forms of discriminant .

We also define ; by the exercise, . This is called the **class number**.

Moreover, we have , because we can take for and for . We call this form the **principal form**.

**Example 10** **(Examples of quadratic forms with , )**

The following discriminants have class number , hence having only the principal form:

- , with form .
- , with form .
- , with form .
- , with form .
- , with form .

This is in fact the complete list when .

**Example 11** **(Examples of quadratic forms with , )**

The following discriminants have class number , hence having only the principal form:

- , with form .
- , with form .
- , with form .
- , with form .
- , with form .
- , with form .
- , with form .
- , with form .

This is in fact the complete list when .

**Example 12** **(More examples of quadratic forms)**

Here are tables for small discriminants with . When we have

- , with forms and .
- , with forms and .
- , with forms and .
- , with forms and .
- , with forms and .
- , with forms and .

As for we have

- , with forms and .
- , with forms and .
- , with forms and .
- , with forms , and .

**Example 13** **(Even More Examples of quadratic forms)**

Here are some more selected examples:

- has forms , and .
- has forms and .
- has forms , and .

We can now connect this to primes as follows. Earlier we played with , and observed that for odd primes , if and only if some *multiple* of is properly represented by .

Our generalization is as follows:

**Theorem 14** **(Primes represented by some quadratic form)**

Let be a discriminant, and let be an odd prime. Then the following are equivalent:

- , i.e. is a quadratic residue modulo .
- The prime is (properly) represented by
*some*reduced quadratic form in .

This generalizes our result for , but note that it uses in an essential way! That is: if , we know is represented by *some* quadratic form of discriminant \dots but only since do we know that this form reduces to .

*Proof:* First assume WLOG that and . Thus , since otherwise this would imply . Then

hence .

The converse direction is amusing: let for integers , . Consider the quadratic form

It is primitive of discriminant and . Now may not be reduced, but that’s fine: just take the reduction of , which must also properly represent .

Thus to every discriminant we can attach the **Legendre character** (is that the name?), which is a homomorphism

with the property that if is a rational prime not dividing , then . This is abuse of notation since I should technically write , but there is no harm done: one can check by quadratic reciprocity that if then . Thus our previous result becomes:

**Theorem 15** **( consists of representable primes)**

Let be prime. Then if and only if some quadratic form in represents .

As a corollary of this, using the fact that one can prove that

**Corollary 16** **(Fermat-type results for )**

Let be a prime. Then is

- of the form if and only if .
- of the form if and only if .
- of the form if and only if .

*Proof:* The congruence conditions are equivalent to , and as before the only point is that the only reduced quadratic form for these is the principal one.

What if ? Sometimes, we can still figure out which primes go where just by taking mods.

Let . Then it **represents** some residue classes of . In that case we call the set of residue classes represented the **genus** of the quadratic form .

**Example 17** **(Genus theory of )**

Consider , with

We consider the two elements of :

- represents .
- represents .

Now suppose for example that . It must be represented by one of these two quadratic forms, but the latter form is never and so it must be the first one. Thus we conclude that

- if and only if .
- if and only if .

The thing that makes this work is that each genus appears exactly once. We are not always so lucky: for example when we have that

**Example 18** **(Genus theory of )**

The two elements of are:

- , which represents exactly the elements of .
- , which
*also*represents exactly the elements of .

So the best we can conclude is that OR if and only if This is because the two distinct quadratic forms of discriminant happen to have the same genus.

We now prove that:

**Theorem 19** **(Genii are cosets of )**

Let be a discriminant and consider the Legendre character .

- The genus of the principal form of discriminant constitutes a subgroup of , which we call the
**principal genus**. - Any genus of a quadratic form in is a coset of the principal genus in .

*Proof:* For the first part, we aim to show is multiplicatively closed. For , we use the fact that

For , we instead appeal to another “magic” identity

and it follows from here that is actually the set of squares in , which is obviously a subgroup.

Now we show that other quadratic forms have genus equal to a coset of the principal genus. For , with we can write

and thus the desired coset is shown to be . As for , we have

so the desired coset is also , since was the set of squares.

Thus every genus is a *coset* of in . Thus:

**Definition 20**

We define the quotient group

which is the set of all genuses in discriminant . One can view this as an abelian group by coset multiplication.

Thus there is a natural map

(The map is surjective by Theorem~14.) We also remark than is quite well-behaved:

**Proposition 21** **(Structure of )**

The group is isomorphic to for some integer .

*Proof:* Observe that contains all the squares of : if is the principal form then . Thus claim each element of has order at most , which implies the result since is a finite abelian group.

In fact, one can compute the order of exactly, but for this post I Will just state the result.

**Theorem 22** **(Order of )**

Let be a discriminant, and let be the number of distinct odd primes which divide . Define by:

- if .
- if and .
- if and .
- if and .
- if and .

Then .

We have already used once the nice identity

We are going to try and generalize this for any two quadratic forms in . Specifically,

**Proposition 23** **(Composition defines a group operation)**

Let . Then there is a unique and bilinear forms for such that

- .
- .
- .

In fact, without the latter two constraints we would instead have and , and each choice of signs would yield one of four (possibly different) forms. So requiring both signs to be positive makes this operation well-defined. (This is why we like proper equivalence; it gives us a well-defined group structure, whereas with improper equivalence it would be impossible to put a group structure on the forms above.)

Taking this for granted, we then have that

**Theorem 24** **(Form class group)**

Let , be a discriminant. Then becomes an abelian group under composition, where

- The identity of is the principal form, and
- The inverse of the form is .

This group is called the **form class group**.

We then have a group homomorphism

Observe that and are inverses and that their images coincide (being improperly equivalent); this is expressed in the fact that has elements of order . As another corollary, the number of elements of with a given genus is always a power of two.

We now define:

**Definition 25**

An integer is **convenient** if the following equivalent conditions hold:

- The principal form is the only reduced form with the principal genus.
- is injective (hence an isomorphism).
- .

Thus we arrive at the following corollary:

**Corollary 26** **(Convenient numbers have nice representations)**

Let be convenient. Then is of the form if and only if lies in the principal genus.

Hence the represent-ability depends only on .

OEIS A000926 lists 65 convenient numbers. This sequence is known to be complete except for at most one more number; moreover the list is complete assuming the Grand Riemann Hypothesis.

To treat the cases where is not convenient, the correct thing to do is develop class field theory. However, we can still make a little bit more progress if we bring higher reciprocity theorems to bear: we’ll handle the cases and , two examples of numbers which are not convenient.

First, we prove that

To do this we use cubic reciprocity, which requires working in the Eisenstein integers where is a cube root of unity. There are six units in (the sixth roots of unity), hence each nonzero number has six **associates** (differing by a unit), and the ring is in fact a PID.

Now if we let be a prime not dividing , and is coprime to , then we can define the **cubic Legendre symbol** by setting

Moreover, we can define a **primary** prime to be one such that ; given any prime exactly one of the six associates is primary. We then have the following reciprocity theorem:

**Theorem 28** **(Cubic reciprocity)**

If and are disjoint primary primes in then

We also have the following supplementary laws: if , then

The first supplementary law is for the unit (analogous to ) while the second reciprocity law handles the prime divisors of (analogous to .)

We can tie this back into as follows. If is a rational prime then it is represented by , and thus we can put for some prime , . Consequently, we have a natural isomorphism

Therefore, we see that a given is a cubic residue if and only if .

In particular, we have the following corollary, which is all we will need:

**Corollary 29** **(When is a cubic residue)**

Let be a rational prime, . Write with primary. Then is a cubic residue modulo if and only if .

*Proof:* By cubic reciprocity:

Now we give the proof of Theorem~27. *Proof:* First assume

Let be primary, noting that . Now clearly , so done by corollary.

For the converse, assume , with primary and . If we set for integers and , then the fact that and is enough to imply that (check it!). Moreover,

as desired.

This time we work in , for which there are four units , . A prime is **primary** if ; every prime not dividing has a unique associate which is primary. Then we can as before define

where is primary, and is nonzero mod . As before , we have that is a quartic residue modulo if and only if thanks to the isomorphism

Now we have

**Theorem 30** **(Quartic reciprocity)**

If and are distinct primary primes in then

We also have supplementary laws that state that if is primary, then

Again, the first law handles units, and the second law handles the prime divisors of . The corollary we care about this time in fact uses only the supplemental laws:

**Corollary 31** **(When is a quartic residue)**

Let be a prime, and put with primary. Then

and in particular is a quartic residue modulo if and only if .

*Proof:* Note that and applying the above. Therefore

Now we assumed is primary. We claim that

Note that since was is divisible by , hence divides . Thus

since is odd and is even. Finally,

From here we quickly deduce

**Theorem 32** **(On )**

If is prime, then if and only if and is a quartic residue modulo .

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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.

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.

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.)

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.

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.

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.

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.

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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.

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.

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.

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.

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!)

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.

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