I think this post is more than two years late in coming, but anywhow…

This post introduces the $p$-adic integers $\mathbb Z_p$, and the $p$-adic numbers $\mathbb Q_p$. The one-sentence description is that these are “integers/rationals carrying full mod $p^e$ information” (and only that information).

The first four sections will cover the founding definitions culminating in a short solution to a USA TST problem.

In this whole post, $p$ is always a prime. Much of this is based off of Chapter 3A from Straight from the Book.

1. Motivation

Before really telling you what $\mathbb Z_p$ and $\mathbb Q_p$ are, let me tell you what you might expect them to do.

In elementary/olympiad number theory, we’re already well-familiar …

I’ve added a new Euclidean geometry handout, Constructing Diagrams, to my webpage.

Some of the stuff covered in this handout:

• Advice for constructing the triangle centers (hint: circumcenter goes first)
• An example of how to rearrange the conditions of a problem and draw a diagram out-of-order
• Some mechanical suggestions such as dealing with phantom points
• Some examples of computer-generated figures

Enjoy.

Some thoughts about some modern trends in mathematical olympiads that may be concerning.

I. The story of the barycentric coordinates

I worry about my geometry book. To explain why, let me tell you a story.

When I was in high school about six years ago, barycentric coordinates were nearly unknown as an olympiad technique. I only heard about it from whispers in the wind from friends who had heard of the technique and thought it might be usable. But at the time, there were nowhere where everything was written down explicitly. I had a handful of formulas online, a few helpful friends I can reach out to, and a couple example posts littered across some forums.

Seduced by the possibility of arcane power, I didn’t let this stop me. Over the spring of 2012, spring break settled in, and I spent that entire week developing the entire theory of …

It’s not uncommon for technical books to include an admonition from the author that readers must do the exercises and problems. I always feel a little peculiar when I read such warnings. Will something bad happen to me if I don’t do the exercises and problems? Of course not. I’ll gain some time, but at the expense of depth of understanding. Sometimes that’s worth it. Sometimes it’s not.

— Michael Nielsen, Neural Networks and Deep Learning

1. Synopsis

I spent the first few days of my recent winter vacation transitioning all the problem sets for my students from a “traditional” format to a “point-based” format. Here’s a before and after.

Technical specification:

• The traditional problem sets used to consist of a list of 6-9 olympiad problems of varying difficulty, for which you were expected to solve all problems over …

1. Synopsis

One of the major headaches of using complex numbers in olympiad geometry problems is dealing with square roots. In particular, it is nontrivial to express the incenter of a triangle inscribed in the unit circle in terms of its vertices.

The following lemma is the standard way to set up the arc midpoints of a triangle. It appears for example as part (a) of Lemma 6.23.

Theorem 1 (Arc midpoint setup for a triangle)

Let $ABC$ be a triangle with circumcircle $\Gamma$ and let $M_A$, $M_B$, $M_C$ denote the arc midpoints of $\widehat{BC}$ opposite $A$, $\widehat{CA}$ opposite $B$, $\widehat{AB}$ opposite $C$.

Suppose we view $\Gamma …$

Median Putnam contestants, willing to devote one of the last Saturdays before final exams to a math test, are likely to receive an advanced degree in the sciences. It is counterproductive on many levels to leave them feeling like total idiots.

— Bruce Reznick, “Some Thoughts on Writing for the Putnam”

Last February I made a big public apology for having caused one of the biggest scoring errors in HMMT history, causing a lot of changes to the list of top individual students. Pleasantly, I got some nice emails from coaches who reminded me that most students and teams do not place highly in the tournament, and at the end of the day the most important thing is that the contestants enjoyed the tournament.

So now I decided I have to apologize for 2016, too.

The story this time is that I inadvertently sent over 100 students home having solved two …

In a previous post I tried to make the point that math olympiads should not be judged by their relevance to research mathematics. In doing so I failed to actually explain why I think math olympiads are a valuable experience for high schoolers, so I want to make amends here.

1. Summary

In high school I used to think that math contests were primarily meant to encourage contestants to study some math that is (much) more interesting than what’s typically shown in high school. While I still think this is one goal, and maybe it still is the primary goal in some people’s minds, I no longer believe this is the primary benefit.

My current belief is that there are two major benefits from math competitions:

1. To build a social network for gifted high school students with similar interests.
2. To provide a challenging experience that lets gifted students …

I recently had a combinatorics paper appear in the EJC. In this post I want to brag a bit by telling the “story” of this paper: what motivated it, how I found the conjecture that I originally did, and the process that eventually led me to the proof, and so on.

This work was part of the Duluth REU 2017, and I thank Joe Gallian for suggesting the problem.

1. Background

Let me begin by formulating the problem as it was given to me. First, here is the definition and notation for a “block-ascending” permutation.

Definition 1. For nonnegative integers $a_1$, …, $a_n$ an $(a_1, \dots, a_n)$-ascending permutation is a permutation on $\{1, 2, \dots, a_1 + \dots + a_n\}$ whose descent set is …

This is a rare politics post; I’ll try to keep this short and emotion-free. If parts of this are wrong, please correct me. More verbose explanations here, here, here, here, longer discussion here.

Suppose you are a math PhD student at MIT. Officially, this “costs” $50K a year in tuition. Fortunately this number is meaningless, because math PhD students serve time as teaching assistants in exchange for having the nominal sticker price waived. MIT then provides a stipend of about $25K a year for these PhD student’s living expenses. This stipend is taxable, but it’s small and you’d pay only $1K-$2K in federal taxes (about 6%).

The new GOP tax proposal strikes 26 U.S. Code 117(d) which would cause the $50K tuition waiver to also become taxable income: the PhD student would pay taxes on an “income” of $75K, at tax brackets of …

I wanted to quickly write this proof up, complete with pictures, so that I won’t forget it again. In this post I’ll give a combinatorial proof (due to Joyal) of the following:

Theorem 1 (Cayley’s Formula)

The number of trees on $n$ labeled vertices is $n^{n-2}$.

Proof: We are going to construct a bijection between

• Functions $\{1, 2, \dots, n\} \rightarrow \{1, 2, \dots, n\}$ (of which there are $n^n$) and
• Trees on $\{1, 2, \dots, n\}$ with two distinguished nodes $A$ and $B$ (possibly $A=B$).

This will imply the answer.

Let’s look at the first piece of data. We can visualize it as $\mathrm{n\; \dots }$