Olympiad Articles
Over time I've written a few handouts pertaining to olympiad math. Here is a list of them. A few are written in traditional Chinese as I was practicing my Chinese in preparation for the Taiwan TST.
The Math and Problem Solving sections of my personal blog might also be of interest. See also Recommendations for other authors I like, as well as my geometry book for a comprehensive textbook in Euclidean geometry. See also Problems for contest papers.
If you notice any errors, please let me know!
LaTeX notes: I provided the LaTeX source for most of these files as an example here. But note to compile these documents in LaTeX, you will need my style file, which can be found here: evan.sty. Because this style file evolves over time, your output might look a little different than the PDF's attached here. For files with Asymptote diagrams you will additionally need olympiad.asy and cse5.asy.
General

English (pdf) (tex)
Notes on proofwriting style. These were used at MOP 2016. 
USAMO 2014 Contest Analysis (pdf) (tex)
This describes in detail the thought process behind each of my solutions to the USAMO 2014. Click here for a copy of the solutions I submitted. 
IMO 2014 Journal (pdf) (tex)
This describes my experiences competing as TWN2 at the 55th IMO 2014.
To download the pictures in the report, locate "media" in the source folder. 
Taiwan TST 2014 Reflection (pdf) (tex)
This describes my experiences competing for a position on the Taiwan IMO 2014 team. It also contains an extensive commentary on each of the Team Selection Tests and Quizzes, which together covered most of the 2013 IMO Shortlist.
Note: Click here for English versions of the TST problems.
Algebra and Number Theory

Introduction to Functional Equations (pdf) (tex)
An introduction to functional equations for olympiad students. 
Monsters (pdf) (tex)
A handout discussing pathological functional equations. 
Summation (pdf) (tex)
General discussion of sums and products, and how to deal with them. Includes generating functions. 
The Chinese Remainder Theorem (pdf) (tex)
An article on the Chinese Remainder "Theorem". 
Orders Modulo a Prime (pdf) (tex)
Article on orders and primitive roots, in particular featuring the sum of squares lemma and its generalization to arbitrary cyclotomic polynomials.
Combinatorics
 Expected Uses of Probability (pdf) (tex)
Inspired by PoShen Loh's talk at the 55th IMO 2014 or Problem 6. Talks about expected value and the probabilistic method. This article was included in Mathematical Reflections, Issue 6 of 2014.
Geometry

Barycentric Coordinates in Olympiad Geometry (full) (abridged)
One of my most famous handouts. Introduces from scratch the method of barycentric coordinates. A cleanedup version of it is found in the draft of my geometry textbook. 
How to Use Directed Angles (pdf) (tex)
A short note on the use of directed angles in olympiad solutions. Why didn't anyone beat me to writing this? 
The Incenter/Excenter Lemma (pdf) (tex)
A collection of problems which exhibit the first olympiad configuration I got to know well, the famous "incenter/excenter lemma". 
Bashing Geometry with Complex Numbers (pdf) (tex)
English translation of my original notes in Chinese. Describes the classic method of complex numbers.
Original Chinese version: (pdf) (tex) 
A Guessing Game: Mixtilinear Incircles (pdf) (tex)
A quick description of some nice properties of mixtilinear incircles. Presented as a "guessing game" where one has to guess collinear points, cyclic quadrilaterals, and so on beforehand. 
Writing Olympiad Geometry Problems (pdf) (tex)
For students who are interested in writing their own olympiad geometry problems! Or more generally, anyone who is curious how my geometry problems get created.
Also: chapter 2 (on power of point) or chapter 8 (on inversion) of my textbook.
Inequalities

Lagrange Multipliers Done Correctly (pdf) (tex)
This is a description of the conditions necessary to execute a Lagrange Multipliers solution on an olympiad. 
SOS: A Dumbass's Perspective (pdf) (tex)
Describes the SOS method for solving inequalities. 
Olympiad Inequalities (pdf) (tex)
English translation of my original notes in Chinese. Describes some "standard strategies" for handling olympiad inequalities.
Original Chinese version: (pdf) (tex)
Miscellaneous

OMO Spring 2014 Executive Report (pdf) (tex)
A short report on running the Online Math Open for Spring 2014. This is mostly for the sake of those who will be running the future installments of the contest, as one of us is graduating. 
Chinese Terminology Sheet (pdf) (tex)
These are the notes that I took when I was studying traditional Chinese in preparation for the Taiwan IMO selection and training camps. It is updated occasionally as I add entries (e.g. when translating HMMT contests). 
LaTeX Example File (pdf) (tex)
A very short document which shows what a "typical" LaTeX file looks like. It was written for a complete beginner. 
USAMO 2003 Rubric (pdf)
The grading rubric for USAMO 2003.