A trailer for p-adic analysis, first half: USA TST 2003

I think this post is more than two years late in coming, but anywhow... This post introduces the $latex {p}&fg=000000$-adic integers $latex {\mathbb Z_p}&fg=000000$, and the $latex {p}&fg=000000$-adic numbers $latex {\mathbb Q_p}&fg=000000$. The one-sentence description is that these are ``integers/rationals carrying full mod $latex {p^e}&fg=000000$ information'' (and only that information). The first four sections will… Continue reading A trailer for p-adic analysis, first half: USA TST 2003

New oly handout: Constructing Diagrams

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… Continue reading New oly handout: Constructing Diagrams

Revisiting arc midpoints in complex numbers

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… Continue reading Revisiting arc midpoints in complex numbers

An apology for HMMT 2016

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… Continue reading An apology for HMMT 2016

A story of block-ascending permutations

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… Continue reading A story of block-ascending permutations

Joyal’s Proof of Cayley’s Tree Formula

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 $latex {n}&fg=000000$ labelled vertices is $latex {n^{n-2}}&fg=000000$. Proof: We are going to construct a… Continue reading Joyal’s Proof of Cayley’s Tree Formula

Positive Definite Quadratic Forms

I'm reading through Primes of the Form $latex {x^2+ny^2}&fg=000000$, by David Cox (link; it's good!). Here are the high-level notes I took on the first chapter, which is about the theory of quadratic forms. (Meta point re blog: I'm probably going to start posting more and more of these more high-level notes/sketches on this blog… Continue reading Positive Definite Quadratic Forms

Some Thoughts on Olympiad Material Design

(This is a bit of a follow-up to the solution reading post last month. Spoiler warnings: USAMO 2014/6, USAMO 2012/2, TSTST 2016/4, and hints for ELMO 2013/1, IMO 2016/2.) I want to say a little about the process which I use to design my olympiad handouts and classes these days (and thus by extension the… Continue reading Some Thoughts on Olympiad Material Design

Holomorphic Logarithms and Roots

In this post we'll make sense of a holomorphic square root and logarithm. Wrote this up because I was surprised how hard it was to find a decent complete explanation. Let $latex {f : U \rightarrow \mathbb C}&fg=000000$ be a holomorphic function. A holomorphic $latex {n}&fg=000000$th root of $latex {f}&fg=000000$ is a function $latex {g… Continue reading Holomorphic Logarithms and Roots

Facts about Lie Groups and Algebras

In Spring 2016 I was taking 18.757 Representations of Lie Algebras. Since I knew next to nothing about either Lie groups or algebras, I was forced to quickly learn about their basic facts and properties. These are the notes that I wrote up accordingly. Proofs of most of these facts can be found in standard… Continue reading Facts about Lie Groups and Algebras