People sometimes ask me, why do we have international students at MOP? Doesn’t that mean we’re training teams from other countries? So I want to make this clear now: the purpose of MOP is not to train and select future IMO teams.

I know it might seem that way, because we invite by score and grade. But I really think the purpose of MOP is to give each one of you the experience of working hard and meeting new people, among other things. Learn math, face challenges, make friends, the usual good stuff, right? And that’s something you can get no matter what your final rank is, or whether you make IMO or EGMO or even next year’s MOP. The MOP community is an extended family, and you are all part of it now.

What I mean to say is, the camp is designed with all 80 of you in mind. It made me sad back in 2012 when one of my friends realized he had little chance of making it back next year, and told me that MAA shouldn’t have invited him to begin with. Even if I can only take six students to the IMO each year, I never forget the other 74 of you are part of MOP too.

This means one important thing: everyone who puts in their best shot deserves to be here. (And unfortunately this also means there are many other people who deserve to be here tonight too, and are not. Maybe they solved one or two fewer problems than you did; or maybe they even solved the same number of problems, but they are in 11th grade and you are in 10th grade.)

Therefore, I hope to see all of you put in your best effort. And I should say this is not easy to do, because MOP is brutal in many ways. The classes are mandatory, we have a 4.5-hour test every two days, and you will be constantly graded. You will likely miss problems that others claim are easy. You might find out you know less than you thought you did, and this can be discouraging. Especially in the last week, when we run the TSTST, many of you will suddenly realize just how strong Team USA is.

So I want to tell you now, stay determined in the face of adversity. This struggle is your own, and we promise it’s worth it, no matter the outcome. We are rooting for you, and your friends sitting around you are too. (And if the people around you aren’t your friends yet, change that asap.)

]]>
As usual, to make these things easier to talk about, I’m going to introduce some words to describe these two. Taking a page from martial arts, I’m going to run with **hard** and **soft** techniques.

A **hard** technique is something you try in the hopes it will prove something — ideally, solve the problem, but at least give you some intermediate lemma. Perhaps a better definition is “things that will end up in the actual proof”. Examples include:

- Angle chasing in geometry, or proving quadrilaterals are cyclic.
- Throwing complex numbers at a geometry problem.
- Plugging in some values into a functional equation (which gives more equations to work with).
- Taking a given Diophantine equation modulo to get some information, or taking -adic evaluations.
- Trying to perform an induction, for example by deleting an element.
- Trying to write down an inequality that when summed cyclically gives the desired conclusion.
- Reducing the problem to one or more equivalent claims.

and so on. I’m sure you can come up with more examples.

In contrast, a **soft** technique is something you might try to help you understand the problem better — even if it might not prove anything. Perhaps a better definition is “things not written up”. Examples include:

- Examining particular small cases of the problem.
- Looking at the equality cases of a min/max problem.
- Considering variants of the problem (for example, adding or deleting conditions).
- Coming up with lots of concrete examples and playing with them.
- Trying to come with a counterexample to the problem’s assertion and seeing what the obstructions are.
- Drawing pictures, even on non-geometry problems (see JMO2 and JMO5 in my 2019 notes for example).
- Deciding whether or not a geometry problem is “purely projective”.
- Counting the algebraic degrees of freedom in a geometry problem.
- Checking all the linear/polynomial solutions to a functional equation, in order to get a guess what the answer might be.
- Blindly trying to guess solutions to an algebraic equation.
- Making up an artificial unnatural function in a functional equation, and then trying to see why it doesn’t work (or occasionally being surprised that it does work).
- Thinking about why a certain hard technique you tried failed, or even better convincing yourself it cannot work (for example, this Diophantine equation has a solution modulo every prime, so stop trying to one-shot by mods).
- Giving a heuristic argument that some claim should be true or false (“probably is odd infinitely often”), or even easy/hard to prove.

and so on. There is some grey area between these two, some of the examples above might be argued to be in the other category (especially in context of specific problems), but hopefully this gives you a sense of what I’m talking about.

If you look at things I wrote back when I was in high school, you’ll see this referred to as “attacking” and “scouting” instead. This is too silly for me now even by my standards, but back then it was because I played a lot of *StarCraft: Brood War* (I’ve since switched to StarCraft II). The analogy there is pretty self-explanatory: knowing what your opponent is doing is important because your army composition and gameplay decisions should change in reaction to more information.

Now after all that blabber, here’s the action item for you all: **you should try soft techniques when stuck**.

When you first start doing a problem, you will often have some good ideas for what to try. (For example: a wild geometry appeared, let’s scout for cyclic quadrilaterals.) Sometimes if you are lucky enough (especially if the problem is easier) this will be enough to topple the problem, and you can move on. But more often what happens is that eventually you run out of steam, and the problem is still standing. When that happens, my advice is to try doing some soft techniques if you haven’t already done so.

Here’s an example that I like to give.

**Example 1** **(USA TST 2009)**

Find all real numbers , , which satisfy

A common first thing that people will try to do is add the first two equations, since that will cause the terms to cancel. This gives a factor of in the left and an in the right, so then maybe you try to submit that into the in the last equation, so you get , cool, there’s no more linear terms. Then. . .

Usually this doesn’t end well. You add this and subtract that and in the end all you see is equation after equation, and after a while you realize you’re not getting anywhere.

So we’re stuck now. What to do? I’ll now bring in two of the soft techniques I mentioned earlier:

- Let’s imagine the problem had replaced with . In this new problem, you can
*imagine*solving for in terms of using the first equation, then in terms of , and then finally putting everything into the last equation to find a degree polynomial in . I say “imagine” because wow would that be ugly.But here’s the kicker: it’s a polynomial. It should have exactly complex roots, with multiplicity. That’s a lot. Really?

So here’s a hint you might take: there’s a good reason this is over but not . Often these kind of things end up being because there’s an inequality going on somewhere, so there will only be a few real solutions even though there might be tons of complex ones.

- Okay, but there’s an even more blatant thing we don’t know yet:
*what is the answer, anyways*?This was more than a little bit embarrassing. We’re half an hour in to the problem and thoroughly stuck, and we don’t even have a single that works? Maybe it’d be a good idea to fix that, like,

*right now*. In the simplest way possible: guess and check.It’s much easier than it sounds, since if you pick a value of , say, then you get from the third equation, from the first, then check whether it fits the second. If we restrict our search to integer values of , then there aren’t so many that are reasonable.

I won’t spoil what the answer is, other than saying there is an integer triple and it’s not hard to find it as I described. Once you have these two meta-considerations, you suddenly have a much better foothold, and it’s not too hard to solve the problem from here (for a USA TST problem anyways).

I pick this example because it really illustrates how hopeless repeatedly using hard techniques can be if you miss the right foothold (and also because in this problem it’s unusually tempting to just think that more manipulation is enough). It’s not *impossible* to solve the problem without first realizing what the answer is, but it is certainly way more difficult.

What this also means is that, in the after-math of a problem (when you’ve solved/given up on a problem and are reading and reflecting on the solution), you should also add soft techniques into the list of possible answers to “how might I have thought of that?”. An example of this is at the end of my earlier post On Reading Solutions, in which I describe how you can come up with solutions to two Putnam problems by thinking carefully about what should be the equality case.

Doing this is harder than it sounds, because the soft techniques are the ones that by definition won’t appear in most written solutions, and many people don’t explicitly even recognize them. But soft techniques are the things that tell you which hard techniques to use, which is why they’re so valuable to learn well.

In writing this post, I’m hoping to make the math contest world more aware that these sorts of non-formalizable ideas are things that can (and should) be acknowledged and discussed, the same way that the hard techniques are. In particular, just as there are a plethora of handouts on every hard technique in the olympiad literature, it should also be possible to design handouts aimed at practicing one or more particular soft techniques.

At MOP every year, I’m starting to see more and more classes to this effect (alongside the usual mix of classes called “inversion” or “graph theory” or “induction” or whatnot). I would love to see more! End speech.

]]>
**tl;dr** I parodied my own book, download the new version here.

People often complain to me about how olympiad geometry is just about knowing a bunch of configurations or theorems. But it recently occurred to me that when you actually get down to its core, the amount of specific knowledge that you need to do well in olympiad geometry is very little. In fact I’m going to come out and say: **I think all the theory of mainstream IMO geometry would not last even a one-semester college course**.

So to stake my claim, and celebrate April Fool’s Day, I decided to **actually do it**. What would olympiad geometry look like if it was taught at a typical college? To find out, I present to you the course notes for:

Undergrad Math 011: a firsT yeaR coursE in geometrY

It’s 36 pages long, title page, preface, and index included. So, there you go. It is also the kind of thing I would never want to read, and the exercises are awful, but what does that matter?

(I initially wanted to post this file as an April Fool’s gag, but became concerned that one would not have to be too gullible to believe these were actual course notes and then attempt to work through them.)

]]>Po-Shen Loh and I spent the last week in Bucharest with the United States team for the 11th RMM. The USA usually sends four students who have not attended a previous IMO or RMM before.

This year’s four students did breathtakingly well:

- Benjamin Qi — gold (rank 2nd)
- Luke Robitaille — silver (rank 10th)
- Carl Schildkraut — gold (rank 8th)
- Daniel Zhu — gold (rank 4th)

(Yes, there are only nine gold medals this year!)

The team score is obtained by summing the three highest scores of the four team members. The USA won the team component by a lofty margin, making it the first time we’ve won back to back. I’m very proud of the team.

RMM 2019 team after the competition (taken by Daniel Zhu’s dad):

McDonald’s trip. Apparently, the USA tradition is that whenever we win an international contest, we have to order chicken mcnuggets. Fortunately, this time we didn’t order one for every point on the team (a silly idea that was unfortunately implemented at IMO 2018).

The winner plate. Each year the winning country brings it back to get it engraved, and returns it to the competition the next year. I will have it for the next while.

And a present from one of the contestants (thanks!):

Amy and Bob play a game. First, Amy writes down a positive integer on a board. Then the players alternate turns, with Bob moving first. On Bob’s turn, he chooses a positive integer and subtracts from the number on the board. On Amy’s turn, she chooses a positive integer and raises the number on the board to the th power. Bob wins if the number on the board ever reads zero. Can Amy prevent Bob from winning?

I found this to be a cute easy problem. The official solution is quite clever, but it’s possible (as I myself did) to have a very explicit solution using e.g. the characterization of which integers are the sum of k squares (for ).

Let be an isosceles trapezoid with . Let be the midpoint of . Denote by and the circumcircles of triangles and , respectively. The tangent to at and the tangent to at intersect at point . Prove that is tangent to .

There are nice synthetic solutions to this problem, but I found it much easier to apply complex numbers with as the unit circle, taking as the phantom point the intersection of tangents. So, unsurprisingly, all our team members solved the problem quite quickly.

I suspect the American students who took RMM Day 1 at home (as part of the USA team selection process) will find the problem quite easy as well. Privately, it is a bit of a relief for me, because if a more difficult geometry had been placed here I would have worried that our team selection this year has (so far) been too geometry-heavy.

Let be a positive real number. Prove that if is sufficiently large, then any simple graph on vertices with at least edges has two (different) cycles of equal length.

A really nice problem with a short, natural problem statement. I’m not good at this kind of problem, but I enjoy it anyways. Incidentally, one of our team members won last year’s IOI, and so this type of problem is right up his alley!

Show that for every positive integer there exists a simple polygon (not necessarily convex) admitting exactly distinct triangulations.

A fun construction problem. I think it’s actually harder than it looks, but with enough time everyone eventually catches on.

Solve over the functional equation

I found this problem surprisingly pernicious. Real functional equations in which all parts of the equation are “wrapped by f” tend to be hard to deal with: one has to think about things like injectivity and the like in order to have any hope of showing that f actually takes on some desired value. And the answer is harder to find that it seems — it is (rightly) worth a point even to get the entire answer correct.

Fortunately for our team members, the rubric for the problem was generous, and it was possible to get 4-5 points without a complete solution. In the USA, olympiad grading tends to be even harsher than in most other countries (though not as Draconian as the Putnam), so this came as a surprise to the team. I jokingly told the team afterwards that they should appreciate how I hold them to a higher standard than the rest of the world.

(Consequently, the statistics for this problem are somewhat misleading — the average score makes the problem seem easier than it actually is. In truth there were not that many 6’s and 7’s given out.)

Find all pairs of integers , both greater than , such that the following holds:

For any monic polynomial of degree with integer coefficients and for any prime , there exists a set of at most integers, such that

contains a complete residue system modulo (i.e., intersects with every residue class modulo ).

Unlike problem 5, I found this problem to be quite straightforward. I think it should have been switched with problem 5. Indeed, all our team members produced complete solutions to this problem.

So I am glad that our team has learned by now to try all three problems seriously on any given day. I have seen too many times students who would spend all their time on a P2 and not solve it, only to find out that P3 was of comparable or easier difficulty.

]]>So since this is someplace between version 1 and the (hopefully eventually) version 2, it seems appropriate to call it **version 1.5**. The biggest changes include a complete rewrite of the algebraic geometry chapters, new parts on real analysis and measure theory, and a reorganization of many of the earlier chapters like group theory and topology, with more examples and problems. There’s also a new chapter 0 entitled “sales pitches” which gives an advertisement for each of the parts later. The obvious gaps: the chapters on probability are yet to be written, as is some more algebraic geometry. The updated flowchart from the beginning of the book is pictured below.

You can download the latest version from the usual page, or directly from https://usamo.files.wordpress.com/2019/02/napkin-v15-20190220.pdf. The number of errors has doubtless increased, and corrections are comments are more than welcome.

Incidentally, this seems as good a time as any to mention two more things:

- My personal website has seen some updates and re-organization, including most notably the OTIS lecture notes that I promised last Christmas, as well as my personal USAMO/IMO solutions.
- Also, I now have a
**public Facebook page**. Right now I mostly plan to use it as a mirror for this blog, but I might also find some other uses for it later. Please feel free to like me ;)

That’s all. Hope you all like it! Best wishes from the Zurich airport.

]]>*That NyQuil stuff really works*, I think to myself, and crawl out of bed. (Even though it’s like trying to drink mouthwash.) I haven’t slept that soundly all week. Or maybe I’m finally slowly recovering from my cold, and that’s why that night was better? All I know is that I’m glad I didn’t spend another night coughing my lungs out and struggling to get some shut-eye.

I drag my sorry butt out of bed and head over to my nearby computer. It’s the Friday morning before Harvard-MIT math tournament, which means that the server is getting more traffic than usual, and I was supposed to have upgraded the server in anticipation yesterday night. But Valentine’s Day was too hectic for me this year, and I never got around to it.

Not hectic for any romantic reasons. It was because I had 5.5 hours of class more or less consecutively, after which I rushed back to my place to teach for another four hours straight, all the while coughing like a banshee. Okay, so maybe I would have slept fine without the NyQuil.

Officially, this is supposed to be the software team’s job. But the HMMT website has become a complete mess that I think I might be the only person left that still knows more than half of what it’s doing. (Well, actually, Banana seems have figured out a lot of it too.) It is sort the equivalent of Frankenstein’s monster, with parts being sewn in and out over the last who-knows-how-many-years by random undergraduates with various degrees of competence, and held together by the seams with spit and prayers. The top of the main settings files still reads

Django settings for mysite project. For more information on this file, see https://docs.djangoproject.com/en/1.6/topics/settings/ For the full list of settings and their values, see https://docs.djangoproject.com/en/1.6/ref/settings/

where the “1.6” version number still makes me wince every time I see it (that means this file was created *before* I made the IMO). I am looking forward to the end of this tournament season so I can burn the whole website to the ground and re-write it.

(Making matters worse, in terms of “various degrees of competence”, I am on the low end, with no formal CS experience at all. Not good I am in charge.)

So it’s time to bump up the servers belatedly. I need to bump the web server and then the database up from t2.micro to m1.medium. There will be some downtime, but no big deal — everyone’s probably still asleep. This should only take a few minutes, and then I can work on getting the grader ready for the approximately 100,000 grading inputs that we’re going to force-feed it on Saturday.

So I push a few buttons, and let Amazon Web Services do its magic, just like the last five times I had to do this.

Something’s wrong. It’s been ten minutes already, and the website still won’t load. The upgrades should be done by now. I refresh again, and realize that it’s throwing a 500 error.

I feel a twinge of despair, which causes my cough to start to return. Looks like the NyQuil wore off. *It always starts like this. One little error, followed by another, and then…* I clench my teeth and SSH into the new server, and navigate to the log files (which I remember having to set up myself a few years ago, precisely for situations like this).

And indeed, when I get there, the same message is repeated, over and over, for the last several minutes, like a harbinger of doom:

OperationalError: (1045, "Access denied for user 'ebroot'@'xx.xx.xx.xx' (using password: YES)")

Oh, well, here we go.

My first guess is VPC groups, which have bitten me plenty in the past. Unfortunately, tinkering with these has no effect, so it seems like there is something new going on.

I double-check and triple-check the password, but it seems right. (This is the same error you get if you enter the wrong password!) So that’s not it, either.

This is a connection issue, so, the first thing I want to do is figure out whether the issue is with the upgraded server or the upgraded database. So, I go to my command bar and launch MySQL workbench, which promptly gives me

mysql-workbench: command not found

Oh, right, I don’t have Workbench installed on my desktop. So I go over to my laptop, which does have it, and try to fire up a connection to the database. This time, it gives me

喊叫 org.freedesktop.secrets 發生錯誤

Uhh, what? Okay, I have a keyring error of some sort here. I decide it’s not worth it to try to fix the workbench, and decide to do it the old fashioned way. I fire another terminal and call mysql, which promptly returns with:

mysql: command not found

Great. Okay, well, I guess I can install it, no problem. So I type in `sudo pacman -S mysql`

which gives

:: 有 2 個提供者可供 mysql：:: 軟體庫 extra1) mariadb:: 軟體庫 community2) percona-server輸入某個數字（預設=1）:

Huh? Oh, Arch Linux prefers MariaDB over MySQL, I remember now. That means SQL, unlike MariaDB, is not pre-packaged, and I’ll need to download it from the Arch User Repository and compile it form source. So I download the PKGBUILD and let it start going.

Unfortunately, this means that the binary needs to build from source, and so maybe minutes later I’m staring at the build progress and it’s at 14%. All the while the website is down, and people are starting to notice. I can feel the tension all over my body as I realize that the tournament will simply not function if I can’t get this back up and working, and the nightmare from February 2018 starts coming back to my mind again. I explode into a symphony of coughs as I struggle to gain my composure.

The compile-from-source doesn’t respond to my please. I decided that’s not good enough, and I have to do something faster.

At this point I decide I maybe should get help, so I hit up Banana who has saved HMMT on numerous occasions as well, and ask him why I can’t connect to the HMMT database. He’s done so successfully on his computer in the past, and he even has a little script he’s written to do so. His response puzzles me:

“Seems connectable. Might need to reset your DNS?”

Uh, what?

This is the only lead I have to go after, so I start prodding him like crazy, since I can’t seem to get it to do anything from my end. At first I get a couple suggestions, which don’t work, but eventually Banana finally gives me:

“Uhhhh. Can you try running db-connect on your machine and going from there? I have to start hauling.”

Right. Unlike me, Banana is actually helping the directors move around the 30 gallons of apple juice and 50 gallons of mango nectar and way-too-many gallons of water that will be used today or tomorrow. So I am really on my own.

At least I now know that it’s on the database end, sorta. I quickly locate db-connect: it’s the mini-script that Banana has been using to do the connection. Maybe this will let me get in and see what’s happening. I type:

./db-connect.sh dev

which promptly gives

mysql: command not found

Ah right, we’re still working on that, huh? The sql compilation is going nowhere fast, and so I have to think of something else.

I think of one possible approach: the workbench won’t work on my laptop, but maybe it’ll work on my desktop? I order pacman to install MySQL Workbench on my desktop too, and after a couple dozen agonizing seconds, the download is all done. To my delight, there is no error about org.freedesktop.secrets, and so I impatiently set everything up and login to find:

ERROR 1045 (28000): Access denied for user 'ebroot'@'xx.xx.xx.xx' (using password: YES)

*Oh, no, no, no.* I explode into another fit of coughs which prevent me from screaming at the monitor in frustration.

At this point, I decide this isn’t worth fighting. I can figure it out another time. For now, *the show must go on*.

That means that if I restore a backup of the old database — reverting it back to how it was yesterday, when everything went totally fine — then I can at least get the computer to work now, and worry about what the error was later.

Unfortunately, this is a painfully slow process. (The way backups work is that they don’t *replace* the existing database. Instead, it creates a brand new database somewhere on the cloud, but that has a copy of the same data as the point in time.) I load the backup, and twitch in agony as it slowly creates a new database from the image, setting everything back to how it was earlier. I hope that’s good enough.

After what feels like an eternity, the database is all set. I change the pointer of my now-working WorkBench to the new database and try to connect to see what happens, only to be greeted with 45 seconds of nothing, followed by a simple error message telling me that the connection failed.

Why? Oh yeah, I didn’t send the VPC for the new database. I do that, exhaling, it should be fine now, and connect to the new restored database, only to find:

ERROR 1045 (28000): Access denied for user 'ebroot'@'xx.xx.xx.xx' (using password: YES)

I practically choke on my own spit, which results in another several seconds of me wheezing like heck.

*Nooooooooooooooooooooooo.*

Okay, I have to fix this now.

The only clue I have is that the database script from Banana still works. But on my computer I can’t for some reason run it.

I go back to Google (which I have been using extensively the whole time), and then after another few minutes of frantic searching, realize that MariaDB is actually good enough for me: once I have that installed, I’ll have a (slightly different) SQL client, but the script should work. Hmm.

Since MariaDB is pre-packaged, that means the installation is easy, and I run it. This might be it. I fire the script again, and run into:

mysql: unknown option '--enable-cleartext-plugin'

Oh, huh. Okay, well, maybe it doesn’t matter. I delete that flag from the script, and get

mysql: unknown variable 'ssl-mode=VERIFY_IDENTITY'

Uhh. Let’s cross our fingers that doesn’t matter either? I try that again, and — much to my amazement — the connection works.

I start examining the db-connect script closely, and see that instead of the user `ebroot`

it’s connecting using some user name `dev`

. So maybe there is some new permission issues with `ebroot`

? With my new connection, I try to

GRANT ALL PRIVILEGES ON *.* TO 'ebroot'@'%';

which then returns with the following message:

ERROR 1045 (28000): Access denied for user 'dev'@'%' (using password: YES)

Uhhh.

Okay, maybe something else. I read the db-connect script again. Is there anything else that’s different? Well, there’s one more change in the code that I can at least work with: there is a CA certificate that’s being used.

I fire up WorkBench again, and try to log in again, but this time I pass a newfound CA certificate (appropriately named *rds-combined-ca-bundle.pem*) and to my relief, I find that I can now log in as ebroot. That’s the issue!

Except I have no idea how to make Django do that and I have no intention if finding out. But another Google search suggests the answer: now that I’m finally connected with ebroot I type

ALTER USER 'ebroot'@'%' REQUIRE NONE;

And breathed a sigh of relief when I refreshed hmmt.co, and harmony was restored in the world.

I looked at the clock. It was 11am. There goes my whole morning. I go downstairs to drink a bottle of Soylent for breakfast, resolving to never become a programmer for a living, or at least to get some proper training first if I ever consider it.

Meanwhile, the rest of the Harvard-MIT math tournament go on with their day, blissfully unaware of the debacle narrowly averted.

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

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

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

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

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

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

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

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

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

For learning theory and fundamentals:

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

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

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

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

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

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

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

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

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

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

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

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

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

You’ll notice how much emphasis I place on the post-problem reflection process. This is actually important — after all the time you spent working on the problem itself, you want to update your instincts as much as possible to get the payoff. In particular, I think it’s usually worth it to read the solutions to problems you worked on, whether or not you solve them. In general, after reading a solution, I think you should be able to state in a couple sentences all the main ideas of the solution, and basically know how to solve the problem from there.

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

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

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

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

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

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

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

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

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

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Applications for OTIS V are open now, so if you are an olympiad contestant interested in working with me during the 2019-2020 school year, here is your chance. I’m hoping to find 20-40 students for the next school year. Note that the application has math problems in it, unlike previous years, so you have to start early.

At the same time, I realize that I will never be able to take everyone for OTIS. So I am planning to post a substantial fraction of OTIS materials for public consumption, hopefully by late January, but no promises.

The Napkin is getting a second edition which, if all goes well, should come out by the end of February (but that is a big “if”). Most chapters will be mostly unchanged modulo typos, but a few big changes:

- I am hoping to add a new part on measure theory with an eye towards probability applications (e.g. law of large numbers, central limit theorem, stopped martingales).
- There will be a bit of real analysis / calculus now. (Not much.)
- Maybe two-ish bonus chapters on other topics being added.
- The earliest chapters (on algebra and topology) are being re-organized significantly, though most of the content should remain the same.
- The algebraic geometry chapters on schemes are getting a major facelift, because the old ones were terrible. They will still cover roughly the same content, but in a way that makes more sense, has more examples, and has more pictures.

This means that for the first time the numbering of the chapters is going to break with the new update. This also means there will be plenty of new typos and mistakes for readers to find. I’m looking forward to it!

For high school students, SPARC applications will open soon. The deadline will probably be the end of February. This year SPARC will be held in the Bay Area from July 24 to August 2.

]]>Here is a link to the working paper.

Despite the click-baity title and dreamy introduction about the Millenium Prizes, the rest of the paper is fascinating, and the figures section is a gold mine. Here are two that stood out to me:

There’s also one really nice idea they had, which was to investigate the effect of getting one point less than a gold medal, versus getting exactly a gold medal. This is a pretty clever way to account for the effect of the prestige of the IMO, since “IMO gold” sounds so much better on a CV than “IMO silver” even though in any given year they may not differ so much. To my surprise, the authors found that “being awarded a better medal appears to have no additional impact on becoming a professional mathematician or future knowledge production”. I included the relevant graph below here.

The data used in the paper spans from IMO 1981 to IMO 2000. This is before the rise of Art of Problem Solving and the Internet (and the IMO was smaller back then, anyways), so I imagine these graphs might look different if we did them in 2040 using IMO 2000 – IMO 2020 data, although I’m not even sure whether I expect the effects to be larger or smaller.

(As usual: I do not mean to suggest that non-IMO participants cannot do well in math later. This is so that I do not get flooded with angry messages like last time.)

]]> Before I go on, I want to mention that is not algebraically closed. So, we can take its algebraic closure — but this field is now no longer complete (in the topological sense). However, we can then take the completion of this space to obtain . In general, completing an algebraically closed field remains algebraically closed, and so there is a larger space which is algebraically closed *and* complete. This space is called the -adic complex numbers.

We won’t need at all in what follows, so you can forget everything you just read.

One of the big surprises of -adic analysis is that we can concretely describe all continuous functions . They are given by a basis of functions

in the following way.

**Theorem 1** **(Mahler; see Schikhof Theorem 51.1 and Exercise 51.B)**

Conversely, if is any sequence converging to zero, then defines a continuous function satisfying (1).

The are called the *Mahler coefficients* of .

**Exercise 2**

Last post we proved that if is continuous and for every then . Re-prove this using Mahler’s theorem, and this time show conversely that a unique such exists when .

You’ll note that these are the same finite differences that one uses on polynomials in high school math contests, which is why they are also called “Mahler differences”.

Thus one can think of as saying that the values of , , \dots behave like a polynomial modulo for every . Amusingly, this fact was used on a USA TST in 2011:

**Exercise 3** **(USA TST 2011/3)**

Let be a prime. We say that a sequence of integers is a *-pod* if for each , there is an such that whenever , divides the sum

Prove that if both sequences and are -pods, then the sequence is a -pod.

We say that a function is **analytic** if it has a power series expansion

As before there is a characterization in terms of the Mahler coefficients:

**Theorem 4** **(Schikhof Theorem 54.4)**

The function is analytic if and only if

Just as holomorphic functions have finitely many zeros, we have the following result on analytic functions on .

**Theorem 5** **(Strassmann’s theorem)**

Let be analytic. Then has finitely many zeros.

We close off with an application of the analyticity results above.

**Theorem 6** **(Skolem-Mahler-Lech)**

Let be an integral linear recurrence. Then the zero set of is eventually periodic.

*Proof:* According to the theory of linear recurrences, there exists a matrix such that we can write as a dot product

Let be a prime not dividing . Let be an integer such that .

Fix any . We will prove that either all the terms

are zero, or at most finitely many of them are. This will conclude the proof.

Let for some integer matrix . We have

Thus we have written in Mahler form. Initially, we define , but by Mahler’s theorem (since ) it follows that extends to a function . Also, we can check that hence is even analytic.

Thus by Strassman’s theorem, is either identically zero, or else it has finitely many zeros, as desired.

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