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

# Category: Mathematics

## Lessons from math olympiads

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… Continue reading Lessons from math olympiads

## 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

## On Reading Solutions

(Ed Note: This was earlier posted under the incorrect title "On Designing Olympiad Training". How I managed to mess that up is a long story involving some incompetence with Python scripts, but this is fixed now.) Spoiler warnings: USAMO 2014/1, and hints for Putnam 2014 A4 and B2. You may want to work on these… Continue reading On Reading Solutions

## 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

## Combinatorial Nullstellensatz and List Coloring

More than six months late, but here are notes from the combinatorial nullsetllensatz talk I gave at the student colloquium at MIT. This was also my term paper for 18.434, ``Seminar in Theoretical Computer Science''. 1. Introducing the choice number One of the most fundamental problems in graph theory is that of a graph coloring,… Continue reading Combinatorial Nullstellensatz and List Coloring