Lessons from Math Olympiads

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

1. Summary

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

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

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

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

2. Social experience

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

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

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

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

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

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

I’ve also heard similar sentiments about ARML:

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

(Don’t tell the parents.)

3. Intellectual growth

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Let me tell you a story.

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

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

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

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

Footnotes

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

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Some Thoughts on Olympiad Material Design

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

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

Now let me explain this in more detail.

1. Main ideas

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

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

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

Example 1 (USAMO 2014, Gabriel Dospinescu)

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

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

If you look at any complete solution to the problem, you will see a lot of technical estimates involving {\zeta(2)} and the like. But the main idea is very simple: “consider an {N \times N} table of primes and note the small primes cannot adequately cover the board, since {\sum p^{-2} < \frac{1}{2}}”. Once you have this main idea the technical estimates are just the grunt work that you force yourself to do if you’re a contestant (and don’t do if you’re retired like me).

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

2. Taxonomy

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

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

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

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

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

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

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

Example 2 (USAMO 2012, Gregory Galperin)

A circle is divided into congruent arcs by {432} points. The points are colored in four colors such that some {108} points are colored Red, some {108} points are colored Green, some {108} points are colored Blue, and the remaining {108} points are colored Yellow. Prove that one can choose three points of each color in such a way that the four triangles formed by the chosen points of the same color are congruent.

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

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

Suppose I wrote down the following:

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

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

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

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

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

3. Ten buckets of geometry

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

  1. Direct angle chasing
  2. Power of a point / radical axis
  3. Homothety, similar triangles, ratios
  4. Recognizing some standard configuration (see Yufei for a list)
  5. Doing some length calculations
  6. Complex numbers
  7. Barycentric coordinates
  8. Inversion
  9. Harmonic bundles or pole/polar and homography
  10. Spiral similarity, Miquel points

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

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

4. How do problems feel?

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

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

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

Example 3 (ELMO 2013, Ray Li)

Let {a_1,a_2,\dots,a_9} be nine real numbers, not necessarily distinct, with average {m}. Let {A} denote the number of triples {1 \le i < j < k \le 9} for which {a_i + a_j + a_k \ge 3m}. What is the minimum possible value of {A}?

Example 4 (IMO 2016)

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

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

If you look at the solutions to these problems, they have the same “feeling” of adding everything up, even though the specific techniques are somewhat different (double-counting for the former, diagonals modulo {3} for the latter). Nonetheless, my experience with problems similar to the former was immensely helpful for the latter, and it’s why I was able to solve the IMO problem.

5. Gaps

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

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

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

6. Surprise

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

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

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

Example 5 (TSTST 2016, Linus Hamilton)

Let {n > 1} be a positive integers. Prove that we must apply the Euler {\varphi} function at least {\log_3 n} times before reaching {1}.

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

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

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

of prime exponents. Then the {\varphi} can be thought of as an operation which takes each nonzero component, decreases it by one, and then adds some particular vector back. For example, if {\nu_7(n) > 0} then {\nu_7} is decreased by one and each of {\nu_2(n)} and {\nu_3(n)} are increased by one. In any case, if you look at this behavior for long enough you will see that the {\nu_2} coordinate is a natural way to “track time” in successive {\varphi} operations; once you figure this out, getting the bound of {\log_3 n} is quite natural. (Details left as exercise to reader.)

Now when I read through the solutions, I found that many of them had not really tried to think of the problem in such a “structured” way, and had tried to directly solve it by for example trying to prove {\varphi(n) \ge n/3} (which is false) or something similar to this. I realized that had the students just ignored the task “prove {n \le 3^k}” and spent some time getting a better understanding of the {\varphi} structure, they would have had a much better chance at solving the problem. Why had I known that structural thinking would be helpful? I couldn’t quite explain it, but it had something to do with the fact that the “main object” of the question was “set in stone”; there was no “degrees of freedom” in it, and it was concrete enough that I felt like I could understand it. Once I understood how multiple {\varphi} operations behaved, the bit about {\log_3 n} almost served as an “answer extraction” mechanism.

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

On Reading Solutions

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

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

1. An Apology

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

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

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

Problem 1 (USAMO 2014)

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

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

can take.

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

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

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

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

Consequently, we have

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

This proves the lower bound. \Box

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

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

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

Unfortunately I have the following confession to make:

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

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

2. Diff

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

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

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

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

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

3. Don’t Read Line by Line

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

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

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

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

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

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

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

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

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

4. Reflection

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

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

So one question you should often ask is:

“How could I have thought of that?”

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

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

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

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

5. Learn Philosophy, not Formalism

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

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

“What does this problem feel like?”

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

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

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

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

Problem 2 (Putnam 2014 A4)

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

Problem 3 (Putnam 2014 B2)

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

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

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

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

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

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

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

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

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