Mathematics Links

Here are a few resources I have developed. Many of these are specifically for math contests, mainly because this is what I spent a good majority of my high school years focused on; hopefully in the future the material here will expand into other areas of mathematics.

Problem Sets

• Homemade Problems (Updated 9/13/2023): This is a collection of (many of) the problems I have submitted and which have appeared on various mathematics competitions throughout the past several years. Solutions are not given, but there are links on the last page that redirect to places where many solutions can be found.

• 100 Geometry Problems: Bridging the Gap from AIME to USA(J)MO: This is a PDF I composed in the summer of 2014, as a project for my 4000th post on Art of Problem Solving. It consists of, as the name suggests, 100 problems, ranging from mid AMC to early USAMO. It seems that the niche for this project has become more crowded as of late with EGMO and Lemmas having been published recently, but I still think it's good for free practice.

• AIME Practice Set 2015 (solutions): This is a set of problems I compiled for AIME practice. The problems here are not from the AIME or AMC - instead, they're from lesser known contests such as HMMT, iTest, and Mandelbrot. They range from early AIME to late AIME in difficulty.

Math League Handouts

From my sophomore year to my senior year of high school, I helped run my high school's math club. This included, among other things, creating and running after-school practices which revolved around single topics. Here are the majority of my handouts from my junior and senior years; these handouts probably have some flaws, but hopefully these might be of use to some of you.

Algebra

• Algebraic Manipulations: A handout which covers the basics of algebraic manipulations problems. The focus is specifically on clever manipulations of systems of equations in order to minimize the amount of brute force needed.

• Sequences and Series: A handout which goes into different types of sequences and series, more specifically arithmetic, geometric, and telescoping.

• Vieta's Formulas: A standard pdf covering Vieta's Formulas in mostly the quadratic and cubic cases.

Combinatorics

• Introduction to Counting and Probability: A handout discussing the three basic methods of combinatorics, namely constructive counting, casework, and complementary counting. The emphasis of this handout is on figuring out which of the three methods to apply in any given situation, as I feel that this is one of the main reasons that combinatorics is so hard.

Geometry

• Angle Chasing: A mini lecture regarding the technique of angle chasing, mostly limited to AMC level problems. The brevity of this PDF comes from the fact that this lecture came after our math club took a Mandelbrot contest.

• Similar Figures: A handout about similarity as it relates to AMC and AIME problems. This assumes that the reader has knowledge of similar triangles already. The focus here is on both nontrivial applications of similar triangles and on consequences of two generic figures being similar (hence the title of the document).

Number Theory

• Introduction to Number Theory: As the name suggests, a handout which goes over some very basic ideas in number theory. While some of the material itself is a bit lacking, the problems I think serve as a good introduction into number theoretic intuition. The sources for these problems are unfortunately commented out in the .tex file; if you're interested in knowing what they are, let me know.

• Diophantine Equations: A very basic overview of some simple techniques (namely factoring) regarding solving Diophantine equations at an AMC level.

• Modular Arithmetic: An introduction to modular arithmetic. The handout starts off with defining residue classes and builds up to more advanced computations. As a warning, it is best to go into this one with a prior knowledge of what modular arithmetic is; although I had inteded this to be an introduction, in reality it builds up way too quickly to be considered as such.

Miscellaneous Trinkets

This section includes handouts or other things which do not fit into the above two frameworks. I expect this section to become much bigger soon.

• Introduction to Conics, Geometrically (new!): An article diving into the basics of a geometric approach to conic sections. This material deserves way more love than it currently gets.

• Sum Uses of Calculus: An article discussing various methods of using calculus techniques to compute various sums. Intended for an undergraduate or advanced high school audience with some familiarity with differentiation and integration.

• Problem Writing Slides: Slides from a talk I gave at AwesomeMath Puget Sound 2019 during the summer. A slightly different version of the talk I gave earlier in the summer (during the Cornell I session) can be found here.

• Conic Geometry: Slides from a talk I gave to CMU students during the summer of 2018. The slides cover a rough introduction to looking at conic sections from a synthetic point of view. The slides are by no means complete, since they were meant to only supplement the talk.

• The Gamma Function: Slides from a talk I gave to CMU students during the winter of 2020 (more specifically, while I was visiting campus to help volunteer for CMIMC 2020).

• Algebraic Topology Notes: Notes from my Algebraic Topology course from the Fall 2018 semester. This is the first time I typeset notes for a course from start to finish, and so these are most definitely quite rough, but I'm happy that I managed to actually finish this massive project, and hopefully I can use the experiences here to make future course notes more polished and more clear.

• Complex Analysis Notes: Notes from my Complex Analysis course from the Fall 2019 semester.