Finally, if you attempt to read this without working through a significant number of exercises (see §0.0.1), I will come to your house and pummel you with [Gr-EGA] until you beg for mercy. It is important to not just have a vague sense of what is true, but to be able to actually get your hands dirty. As Mark Kisin has said, “You can wave your hands all you want, but it still won’t make you fly.”

— Ravi Vakil, The Rising Sea: Foundations of Algebraic Geometry

When people learn new areas in higher math, they are usually required to do some exercises. I think no one really disputes this: you have to actually do math to make any progress.

However, from the teacher’s side, I want to make the case that there is some art to picking exercises, too. In the process of writing my Napkin …

I’m happy to announce that sign-ups for my new olympiad style contest, the United States Ersatz Math Olympiad (USEMO), are open now! The webpage for the USEMO is https://web.evanchen.cc/usemo.html (where sign-ups are posted).

The US Ersatz Math Olympiad is a proof-based competition open to all US middle and high school students. Like many competitions, its goals are to develop interest and ability in mathematics (rather than measure it). However, it is one of few proof-based contests open to all US middle and high school students. You can see more about the goals of this contest in the mission statement.

The contest will run over Memorial day weekend:

• Day 1 is Saturday May 23 2020, from 12:30pm ET – 5:00pm ET.
• Day 2 is Sunday May 24 2020, from 12:30pm ET – 5:00pm ET.

In the future, assuming continued interest …

This post will mostly be focused on construction-type problems in which you’re asked to construct something satisfying property $P$.

Minor spoilers for USAMO 2011/4, IMO 2014/5.

1. What is a leap of faith?

Usually, a good thing to do whenever you can is to make “safe moves” which are implied by the property $P$. Here’s a simple example.

Example 1 (USAMO 2011)

Find an integer $n$ such that the remainder when $2^n$ is divided by $n$ is odd.

It is easy to see, for example, that $n$ itself must be odd for this to be true, and so we can make our life easier without incurring any worries by restricting our search to odd $n$. You might therefore call this an “optimization”: a kind of move that makes the …

I’m now going to say something explicitly that I hinted at in June: I don’t think a student deserves to make MOP more because they had a higher score than another student.

I think it’s easy to get this impression because the selection for MOP is done by score cutoffs. So it sure looks that way.

But I don’t think MOP admissions (or contests in general) are meant to be a form of judgment. My primary agenda is to run a summer program that is good for its participants, and we get funding for N of them. For that, it’s not important which N students make it, as long as they are enthusiastic and adequately prepared. (Admittedly, for a camp like MOP, “adequately prepared” is a tall order). If anything, what I would hope to select for is the people who would get the most …

There’s a new addition to my olympiad problems and solutions archive: I created an index of many past IMO/USAMO/USA TST(ST) problems by what my opinions on their difficulties are. You can grab the direct link to the file below:

https://evanchen.cc/upload/MOHS-hardness.pdf

In short, the scale runs from 0M to 50M in increments of 5M, and every USAMO / IMO problem on my archive now has a rating too.

My hope is that this can be useful in a couple ways. One is that I hope it’s a nice reference for students, so that they can better make choices about what practice problems would be most useful for them to work on. The other is that the hardness scale contains a very long discussion about how I judge the difficulty of problems. While this is my own personal opinion, obviously, I hope it …

Math must be presented for System 1 to absorb and only incidentally for System 2 to verify.

I finally have a sort-of formalizable guideline for teaching and writing math, and what it means to “understand” math. I’ve been unconsciously following this for years and only now managed to write down explicitly what it is that I’ve been doing.

(This post is written from a math-centric perspective, because that’s the domain where my concrete object-level examples from. But I suspect much of it applies to communicating hard ideas in general.)

S1 and S2

The quote above refers to the System 1 and System 2 framework from Thinking, Fast and Slow. Roughly it divides the brain’s thoughts into two categories:

• S1 is the part of the brain characterized by fast, intuitive, automatic, instinctive, emotional responses, For example, when you read the text “2+2=?”, S1 tells you (without …

Up to now I always felt a little saddened when I see people drop out of the IMO or EGMO team selection. But actually, really I should be asking myself what I (as a coach) could do better to make sure the students know we value their effort, even if they ultimately don’t make the team.

Because we sure do an awful job of being supportive of the students, or, well, really doing anything at all. There’s no practice material, no encouragement, or actually no form of contact whatsoever. Just three unreasonably hard problems each month, followed by a score report about a week later, starting in December and dragging in to April.

One of a teacher’s important jobs is to encourage their students. And even though we get the best students in the USA, probably we shouldn’t skip that step entirely, especially given the level …

Here is my commentary for the 2019 International Math Olympiad, consisting of pictures and some political statements about the problem.

Summary

This year’s USA delegation consisted of leader Po-Shen Loh and deputy leader Yang Liu. The USA scored 227 points, tying for first place with China. For context, that is missing a total of four problems across all students, which is actually kind of insane. All six students got gold medals, and two have perfect scores.

1. Vincent Huang 7 7 3 7 7 7
2. Luke Robitaille 7 6 2 7 7 6
3. Colin Shanmo Tang 7 7 7 7 7 7
4. Edward Wan 7 6 0 7 7 7
5. Brandon Wang 7 7 7 7 7 1
6. Daniel Zhu 7 7 7 7 7 7

Korea was 3rd place with 226 points, just one point shy of first, but way ahead of the 4th place score (with 187 points …

While making preparations for this year’s MOP, I imagined to myself what I would say on orientation night if I was director of the camp, and came up with the following speech. I thought it might be nice to share on this blog. Of course, it represents my own views, not the actual views of MOP or MAA. And since I am not actually director of MOP, the speech was never given.

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 …

In yet another contest-based post, I want to distinguish between two types of thinking: things that could help you solve a problem, and things that could help you understand the problem better. Then I’ll talk a little about how you can use the latter. (I’ve talked about this in my own classes for a while by now, but only recently realized I’ve never gotten the whole thing in writing. So here goes.)

1. More silly terminology

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 …