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 …

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 …

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 …

So I have an FAQ now for contest-studying advice, but there’s a “frequently used answer” that I want to document now that doesn’t fit in the FAQ format because the question looks different to everyone that asks it.

The questions generally have the same shape: “would it be better to do X or Y when studying?”. Like:

• Is it better to use GeoGebra when practicing geometry?
• Should I work on some new OTIS units or go back through some old ones that I didn’t finish?
• Should I work on hard problems in my strongest subject or medium problems in my weaker subjects?
• Would it be better if I learned this or that first?

and things like this.

And the answer is, for a lot of pairs (X,Y), if you’re so unsure that you’re asking me about it, then you should just do whatever you …

Updated version of generic advice post: Platitudes v3.

I think this is an important question to answer, not the least of reasons being that understanding how to learn is extremely useful both for teaching and learning.The least of reasons is that people ask me this all the time and I should properly prepare a single generic response.

About a year agoIt’s only been a year? I could have sworn it was two or three., I posted my thoughts on what the most important things were in math contest training. Now that I’m done with the IMO I felt I should probably revisit what I had written.

It looks like the main point of my post a year ago was mainly to debunk the idea that specific resources are important. Someone else phrased this pretty well in the replies to the thread

The issue is many people …