Handouts

Here are all of my handouts, categorized by subject. You can view their LaTeX source files here.

You can also view handouts categorized by whether or not they’re for competition math. If you’re looking to prepare for math contests, you should also look at PRIME.

Algebra

Parallel Sum

A handout about a little-known operator called parallel, its algebraic properties, and its appearances in geometry.

Orbits

A handout about orbits and functional graphs, which some people call thinking about “arrows”.

Equality Cases

A handout about turning inequalities to equalities, by adding them together, using monotonicity, or using equality cases.

Continuants

Talks about some neat facts about continuants, a series of polynomials related to continued fractions.

Newton Interpolation and the Umbral Calculus

An explanation of Newton interpolation that I find makes it easy to remember, and its relation to the calculus of finite differences and umbral calculus in general.

Canonical Decomposition and the First Isomorphism Theorem

The canonical decomposition of a function, as explained in Aluffi’s Algebra Chapter 0, and what I think is the correct way to explain quotienting and the first isomorphism theorem.

Convex Combinations

How I think about convex combinations, both algebraically and geometrically, why they’re named convex combinations, and their relation to Jensen’s inequality.

Combinatorics

Guessing Games

A handout about olympiad problems that have someone asking queries to try to learn information.

Chains and Antichains

A handout about chains and antichains in posets, and how to use Mirsky’s and Dilworth’s theorems on them.

Incidence Matrices

A handout about using incidence matrices to solve set combinatorics problems, often using double counting.

What Is n Choose 2?

A double counting handout specifically themed around counting pairs of things.

Graph Theory

A handout for an introductory graph theory lesson I gave in the Philippine Math Olympiad Summer Camp.

Hall’s Marriage Theorem

Discusses example problems, followed by a problem set. Some basic knowledge about graph theory assumed.

Crossing Numbers

Talks about the Euler characteristic, Kuratowski’s and Wagner’s theorems, Fáry’s theorem, and the crossing number inequality. No prerequisites, but knowledge of graph theory is useful.

Geometry

Parallel Sum

A handout about a little-known operator called parallel, its algebraic properties, and its appearances in geometry.

Compass and Ruler

A light-hearted problem set on compass-and-ruler constructions, as well as constructions with other tools. Best not to take it too seriously.

Synthetic Trigonometry

A problem set on deriving the values of trigonometric functions for common angles using synthetic methods. Helpful for memorizing sin 36° and cos 75°, for example.

A problem set about angle chasing, for a class I gave in the Philippine Math Olympiad Summer Camp. Has corresponding lecture slides. Heavily based on Chapter 1 of Euclidean Geometry for Mathematical Olympiads.

Obscure Geometry Theorems

Discusses some techniques in computational geometry through the proofs of some lesser-known theorems.

Constructions

An experimental problem set on geometry problems that become easier after constructing something.

Pascal’s Theorem

Briskly sets up the projective plane and discusses example problems using Pascal’s theorem, followed by a problem set.

Number Theory

Sledgehammers

A handout about using overpowered NT theorems for olympiad problems.

Four-Function Primality Testing

Here’s how to use a four-function calculator to check that 123,456,789,011 is prime in fifteen minutes. Discusses the Fermat test and the Miller-Rabin test.

Nineteen Proofs There Are Infinitely Many Primes

Nineteen sketches of different proofs there are infinitely primes, each shortened to be readable in about nineteen seconds each.

Lifting the Exponent

A very short handout on lifting the exponent, followed by some problems.

Other

Every Reference in "Finite Simple Group (of Order Two)"

Notes for a Summer HSSP 2022 class aimed at high schoolers. Its ambitious goal is to cover every reference used in the Klein Four’s song, “Finite Simple Group (of Order Two)”.

Type Theory by Example

Type theory, focusing on the theory behind proof assistants, with an emphasis on examples. Prerequisites are some familiarity with logic.

Curry-Howard by Example

Leads up to a specific example of the Curry–Howard correspondence, a neat connection between logic and type theory.

Machine Learning

Machine learning, for people more used to reading math textbooks than writing code. Prerequisites are some knowledge of probability theory, and dealing with matrices and vectors.

Engineering

Engineering is finding the answer to a problem without solving it. People aren’t trying to engineer enough, so I wrote this to change that.

Mathematical Investigations

An article on mathematical investigations. The intended audience are Philippine high school students, in view of the planned curriculum change.

2018–19 in Review

My top ten favorite problems from Philippine high school math contests in 2018–19, and a review of the problems in each contest. Has some insight in setting problems for competitions.

Spelling Out Numbers

A recreational problem set on numbers and their spelled out forms.

Talasalitaang Pangsipnayan

An English-Filipino mathematics glossary I compiled, aimed towards words used for math competitions. Mostly based on Maugnaying Talasalitaang Pang-Agham, edited by Gonasalo del Rosario.

Non-Standard MMC Problems

Practice questions for the oral rounds, compiled for use of our math team. Has actual MMC questions and questions similar in format, mostly from NIMO.

Tiny Explanations

This is a series of self-contained articles, each meant to be about two to three pages long, about random small topics that I wished someone explained to me earlier.