Cubics

Last summer I taught a lecture at MOP about curves of degree higher than 2. I finally posted the handout from that lecture on my website, because a lot of students have been asking me repeatedly, and one of my students went so far as to start a petition.

I am of course flattered and grateful for the enthusiasm. But there was a reason I didn’t post it.

Theater

That reason comes down to the following quote from my 2023 post:

You’ll notice that I’ve started using words like “delivery” or “performance”, and that’s deliberate. To do lectures well, you need to rehearse things like not saying “um”, choosing really flashy or shiny topics, managing blackboard space, keeping an audience engaged, and cutting out content to streamline the presentation (whereas in writing, you can e.g. put digressions …

For high school math olympiad students, OTIS XII applications are out, due August 1, 2026.

But I’m really going to use that short announcement as an excuse to describe the second most common error I see on the OTIS application (the most common error is this type-error I wrote about in 2023). The mistake is so obvious I feel stupid writing about it, but I keep seeing it. So I’m going to record it anywayFree blog advertisement every time I see this mistake again, right?, even though most of the readers of the blog will laugh at me.

Here’s a problem I made up to demonstrate it:

Problem: For real numbers $a$ and $b$, determine the minimum possible value of $$f(a,b) = (ab-1)^2 + (a-3)^2 + (b-3)^2 …$$

Here’s a section from the H-group Hanabi strategy page:

LINES

During your turn, part of figuring out the best move involves looking into the future to see what the next player will do. If they discard, will it be okay? Is there some obvious clue that they will do? And so on.

As you get better at Hanabi, you will need to do this prediction not just for the next player, but for an entire go-around of the table. And as you really get good at Hanabi, you will need to do this for as far in the future as you can reasonably predict. (Sometimes, this means 15 moves or more in the future.)

Similar to chess, initiating a move in which you can predict the next sequence of moves is called initiating a “line”.

In post-game reviews, we will often compare and hypothetically “play through” two different lines …

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 was a coordinator for last year’s IMO 2024 and this year’s IMO 2025.Before, I was a coordinator for some virtual IMO during the pandemic too, which is much less fun. And from 2017-2019 I was an observer for the USA. Here’s some thoughts about that, contrasting my IMO 2019 post.

What is coordination?

For those of you that don’t know, coordination is the grading process for IMO. As I describe it in my FAQ:

Basically, the outline of the idea is: before the exam, a marking scheme (rubric) is set for each problem, to cover the typical cases of what progress will be worth what points. Then, the leaders of each country get to see the solutions of their country’s students, while there is a number of coordinators from the IMO host country for each problem. Both the coordinators and the leaders read …

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 …

The application

During this year’s MOP, we used the following procedure to divide some of our students into two classes:

Let $p = 7075374838595186541578161$ be prime. Take the letters in your name as it appears on the roster, convert them with A1Z26 and take the sum of cubes to get a number $s$. For example, EVANCHEN corresponds to $s = 5^3 + 22^3 + \dots + 14^3 = 16926$. Then you’re in Red 1 (room A155) if $s$ is a quadratic residue modulo $p$, and Red 2 (room A133) otherwise.

The students were understandably a bit confused why the prime was chosen. It turned out to be a prank: if you ran the calculation on the 30-ish students in this class, it was …

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 …

This is a short blog post on the FrontierMath benchmark, a set of lots of difficult math problems with easily verifiable answers. Just to be clear, everything written here is my own thoughts and doesn’t necessarily reflect the intention of any collaborators.

When you’re setting a problem for a competition like the IMO or Putnam, three properties that are often considered desirable are:

1. It should require creative insight. Competitions avoid problems that are too similar to existing ones or too easily solved by simply applying standard textbook techniques. You want the problems to really feel different and force the solver to feel like they came up with a new idea to solve it. This is sort of what the spirit of math olympiads is about.

2. It should not take a lot of implementation, i.e. once a set of key ideas has been identified, actually carrying out the …

There’s this pet peeve I have where people sometimes ask things like what kind of strategies they should use for, say, collinearity problems in geometry.

Like, I know there are valid answers like Menelaus or something. But the reason it bugs me is because “the problem says to prove collinearity” is about as superficial as it gets. It would be like asking for advice for problems that have “ABC” in them.

To drive my point, consider the following setup:

Let $ABC$ be a triangle with circumcircle $\Gamma$ and incenter $I$ and let $M$ be the midpoint of $\overline{BC}$. Denote by $D$ the foot of the perpendicular from $I$ to $\overline{BC}$. The line through $I$ perpendicular to $\overline{AI}$