I had a student at MOP ask me something equivalent to “how should I study while at MOP?”There is also a question about whether you should be studying much at MOP at all — you could also spend a lot of time making new friends, for example. That’s a value judgment that I think is better left to individuals and I won’t comment on it further in this post. For those of you that don’t know, MOP is the three-week summer camp for the USA’s team to the IMO.

At first I was going to just link my FAQ. But then I thought about it a bit more, and I was surprised to find that my answer was not the same as the general how-to-study FAQ. The additional condition “while at MOP” was enough to cause me to stay up that night writing an entirely different …

In my last semester of MIT I led a recitation (i.e. twice-a-week review) sessionFor those of you that don’t know how the system works, at MIT, 18.02 is a huge class with 400 to 500 students (mostly first-years). In order to make sure students actually get the individual attention they need (impossible during lecture), the math department also places each student in a recitation section of about 20 students each, meeting twice a week for an hour each. for multivariable calculus (18.02) at MIT (although the first few weeks are all linear algebra). It’s different from many contexts I’ve taught in before; the emphasis of the class is on doing standard procedures, but the challenge is that there is a lot of ground covered. That is, compared to other settings I’ve taught, there is generally a tradeoff of less depth for more …

I am always harping on my students to write solutions well rather than aiming for just mathematically correct, and now I have a pair of problems to illustrate why.

Shortlist 2011 N1

Here is Shortlist 2011 N1, proposed by Suhaimi Ramly:

For any integer $d > 0$, let $f(d)$ be the smallest positive integer that has exactly $d$ positive divisors (for example, $f(1)=1$, $f(5)=16,$ and $f(6)=12$). Prove that for every integer $k \geq 0$, $f(2^k)$ divides $f(2^{k+1})$.

I like this problem, so try it out if you haven’t. This is a problem …

I recently had a student writing to me asking for advice on problem-solving. The student gave a few examples of problems they didn’t solve (like I tell people to). One of the things that struck me about the message was their description of their work on USAMO 2021/4, whose statement reads:

A finite set $S$ of positive integers has the property that, for each $s\in S$, and each positive integer divisor $d$ of $s$, there exists a unique element $t\in S$ satisfying $\gcd(s,t) = d$. (The elements $s$ and $t$ could be equal.) Given this information, find all possible values for the number of elements of $S$.

Roughly (for privacy reasons, this isn’t exactly what …

There are a lot of different kinds of math enrichment activities now, ranging from olympiads to math circles to tons of summer programs and so on. I work in the competition sphere, and I used to spend a lot of time worrying about whether I took the right side.

Now that I’m a bit older, I came to the realization that maybe I don’t need to be so intent on comparing my work to others (even though I realize comparing yourself to others is human nature, haha). I eventually told myself: there are lots of people who don’t like olympiad exams; there are also lots of people who do, and it’s just okay for them to co-exist. We don’t need to decide which of the N systems is the best and kill the other N-1, because “best” is so different from person to person anyway …

About five years ago I wrote a blog post warning that I thought it was a bad idea to design math olympiads to be completely untrainable, because I think math olympiads should be about talent development rather than just talent identification, yada yada yada.

So now I want to say the other direction: I also don’t want to design math olympiads so that every problem is 100% required to lie in a fixed, rigid, and arbitrary boundary prescribed by some nonexistent syllabus. From a coach’s perspective, I want to reward “good” studying, and whatever “good” means, I think it should include more than zero flexibility and capacity to deal with slight curveballs.

I was reminded of this because there was a recent contest problem (I won’t say which one to avoid spoilers) that quoted Brianchon’s theorem. Brianchon’s theorem, for those of you that don’t …

Sometimes I get asked broad advice questions on solving problems, for example questions like:

1. How do I know when to switch or prioritize approaches I come up with?
2. How do I know which points or lines to add in geometry problems?
3. How can I tell if I’m making progress on a problem?
4. How can I guess the answer if “find all” or “find min/max” problems?
5. How can I tell whether a conjecture I made is true or not?
6. What should I do on a problem when I am stuck?

and so on.

I think all of these questions have a certain quality that, for lack of a better name, I’ll dub as being “NP-hard”. This is a bit of abuse of terminology borrowed from complexity theory, but let me explain why I think the name fits.

We know that solving math problems is generally difficult. There’s …

Editorial note: this post was mostly written in February 2023. Any resemblance to contests after that date is therefore coincidental.

Background

A long time ago, rubrics for the IMO and USAMO were fairly strict. Out of seven, the overall meta-rubric looks like:

• 7: Problem solved
• 6: Tiny slip (and contestant could repair)
• 5: Small gap or mistake, but non-central
• 2: Lots of genuine progress
• 1: Significant non-trivial progress
• 0: “Busy work”, special cases, lots of writing

In particular, traditional rubrics were often sublinear. You’d see problems where you could split it into two parts, and solving either part would only give 2 points, whereas solving both was worth 7.

Increasingly, I’ve noticed this is less and less common. Particularly, at the IMOAs far as I know, the IMO rubrics aren’t really available anywhere. (On the other hand, I’ve never been told that rubrics explicitly need …

I remember reading a Paul Graham essay about how people can’t think clearly about parts of their identity. In my students, I have never seen this more clearly than when people argue about the difficulty of problems.

Some years ago I published a chart of my ratings of problem difficulty, using a scale called MOHS. When I wrote this I had two goals in mind. One was that I thought the name “MOHS” for a Math Olympiad Hardness Scale was the best pun of all time, because there’s a geological scale of mineral hardness that coincidentally has the same name. The other was that I thought it would be useful for beginner students, and coaches, to help find problems that are suitable for practice.

I think it did accomplish those goals. The problem is that I also inadvertently helped catalyze an endless, incessant stream of students constantly arguing …

Note: if you are a prospective OTIS student, read the syllabus instead. More useful, less bragging.

In the unlikely event that I’m a social gathering like a party or family gathering, people will sometimes ask me about my teaching. Invariably they ask, “so do you do like 1:1 meetings or group lessons?”. Then I have to explain, no, I have 400 students, there are no synchronous meetings at all. The core of the program is literally a Python web server that serves PDF files.

Then it sounds less impressive. I guess when people hear I’m a teacher, they expect me to teach classes, and it’s a bit embarrassing to explain that I’m not a teacher in that sense anymore.

But the purpose of OTIS isn’t to make Evan sound cool at parties; the purpose of OTIS to be effective for the students. So this …