This list is sorted by olympiad category, then alphabetically by name.
Each topic has one or more units which is designated by a three-letter code.
This page was last updated Tue, 13 Aug 2019 12:36:44 -0500.
Problems involving some real analysis. This is a pretty technical lecture and will delve into the nuances of converge issues, absolute versus conditional convergence, using calculus properly, compact sets, and so on. Featuring the art of Lagrange multipliers.
Based on Yang Liu's class "Write Down Formulas" at MOP 2018. This unit consists of problems which involve manipulations of fairly involved formulas. Despite officially being algebra, there are just as many (maybe more) problems that would be classified as C or N.
Generating functions and their use in algebraic contexts (computing sums, featuring the so-called snake oil method. Of course, some combinatorial problems included as well.
A unit featuring some of the hardest gems from the golden age of inequalities.
The basics of inequalities: AM-GM, homogenization, Cauchy/Holder.
Techinques for inequalities of the form f(x_1) + ... + f(x_n). Jensen, Karamata, tangent line trick, and n-1 equal value principle.
Inequalities which can be approached using all the standard methods. This is sort of a combination of Ineq Basic and Ineq Func.
Problems about showing polynomials are irreducible; these are rare in olympiads, but give you a lot of good intuition about how Z[x] polynomials behave. Techniques that appear include working in F_p[x], looking at the magnitude of complex roots a la Rouche, and other ad-hoc tricks. More olympiad algebra than the integer polynomials unit.
An extension of the Sums unit --- rather than just swapping infinite sums, we now get to enjoy swapping infinite integrals as well.
Practice with manipulating sums, and in particular switching the order of summation (or integration). Features generating functions and Snake Oil as well.
Vieta formulas, Newton sums, and the fundamental theorem of symmetric polynomials. Involves some computational problems.
Harder inequality problems that don't succumb to the standard methods: it takes some more ingenuity to figure out how to approach these.
Some selected problems revolving around the idea that a function from a set to itself can be thought of as a directed graph with all outdegrees equal to 1. In particular, iterating such a function often involves looking at its cycle decomposition.
A challenging combinatorics unit to conclude the year, with difficult problems reviewing everything that has appeared earlier.
A silly unit on using combinatorial nullstellensatz. Mostly for fun.
An important unit about taking advantage of the equality case in combinatorial problems in order to solve problems. Mandatory for newcomer students.
Computational problems involving probability, expected value (in particular linearity of expectation), and Markov chains (processes which move from state to state).
A difficult unit on problems from extremal graph theory; finding graphs which maximize X under certain constructions. The most basic example is Turan's theorem, which maximizes the number of edges in a graph avoiding an r-clique. Global/local ideas as well as understanding of equality cases feature prominently. Most of the examples in this unit are more challenging.
DCW DCX ZCW ZCX].
Problems for which you have a lot of room to make decisions; a lot of the problems in this unit are constructions, for example. You will feel like you are inventing mathematics, rather than discovering it (in contrast to the Rigid unit).
BCW DCW DCX].
Linearity of expectation, switching the order of summation, what's often called pigeonhole principle, counting in two ways, ... turns out they're actually all more or less the same idea.
Global and Local
An accelerated version of both the global and local units (both done at once).
Combinatorics practice with graphs.
A fun but difficult unit on combinatorics problems involving rectangular grids.
Hall's marriage lemma.
BCX BCY DCX DCY].
Problems which really use induction and recursion is a substantial way, i.e. the main idea of the problem really is about how (or whether) to induct. Features some AIME-style recursion calculations.
Problems using linear algebra (rather than problems about linear algebra). Most of the problems here are combinatorial in nature as a result, and there is a mix of linear algebra over R and linear algebra over F_2.
In contrast to the global unit, this unit is about problems starting from somewhere and perturbing it by a little bit. For example, in a greedy algorithm, if I want a set S of size at least 100 with a certain property, I can imagine starting with S empty and then grabbing things to add to S while trying to avoid bad-ness (whatever that means for the current problem). It's then enough to prove I don't get stuck at any point. Most of the problems in this unit will have a similar algorithmic feeling.
This one's a secret. It's a bit weird, but for this unit to work I have to start by not telling you what it's about.
This is about staring at a moving process (e.g. windmill) and trying to understand what is going on. (The most common thing people say here is monovariants or invariants, but that's only one example of a way you can understand a process.) In a lot of ways it's like the Rigid unit, except your data is way less concrete, and in some cases unobtainable, so you'll be applying the same intuition in a more hostile environment.
DCW DCX ZCW ZCX].
This is one of my favorite units. It's about problems which involve taking a fixed structure, and trying to figure out as much as you can about it --- the task the problem asks you to actually prove becomes unimportant, almost like an answer extraction at the end. Rigid problems often have so few degrees of freedom that a lot of what you'll be doing is writing down a lot of concrete examples, and then trying to figure out what they have in common. You will feel like you are discovering mathematics, rather than inventing it (in contrast to the Free unit).
A fun unit involving combinatorics problems from Russia.
BFW DFW ZFW DFX ZFX].
Functional equations, I guess.
The functional equations that don't bore me, because the solution isn't just f(x) = x anymore! This follows up the Monsters handout on my website.
Wrapped Func Eqn
On real-valued functional equations in which all variables are "wrapped" by the function in some way.
Computational geometry problems, many taken from the tail end of the AIME. At the border of computational contests and olympiads.
DGX ZGX DGY ZGY].
A unit on geometry problems with a highly traditional or synthetic flavor, for example USAMO 2016/3 and USAMO 2017/3. These sorts of problems have recently gained popularity on the USA olympiads and team selection tests and it is totally not my fault. A lot of these problems involve figuring out what certain points are, adding in new points that were not that already, and altogether slowly piecing together a master diagram that reveals the depth of a configuration.
BGW DGW DGX].
Barycentric coordinates in olympiad geometry. A follow-up to Chapter 7 of EGMO.
BGW DGW DGX].
Complex numbers in olympiad geometry. A follow-up to Chapter 6 of EGMO.
BGW DGW BGY DGY].
A unit featuring easy to medium geometry problems which can be solved using only the most basic tools: angle chasing, power of a point, homothety. It can be thought of as a follow-up to Part I of EGMO.
DGW DGX ZGW].
Your friendly projective geometry unit. Harmonic bundles, poles and polars, and so on. A follow-up to Chapter 9 of EGMO.
A less friendly and more abstract projective geometry unit, with an emphasis on projective transformations. Most of the theorems will be stated with respect to an arbitrary conic rather than a circle.
A silly (but difficult) unit on the theory of rectangular circumhyperbolas and the Poncelet point. For fun, if you really like hardcore projective geometry.
Inversion and Spiral
A harder follow-up unit to chapters 8 and 10 of EGMO.
Inversion in olympiad geometry, following up chapter 8 of EGMO.
A technique involving animating points and considering resulting projective maps. This lecture was contributed by Anant Mudgal.
Spiral similarity and Miquel points, following up chapter 10 of EGMO.
A more difficult version of the barycentric coordinates unit.
A more difficult version of the complex numbers unit.
Those weird geometry problems that involve pentagons and hexagons and whatnot (see USAMO 2011/3 for example). Careful use of complex numbers and counting degrees of freedom are important for this unit.
Anti Probs Trilogy
DMW DMX DMY].
A unit consisting entirely of troll hit/miss problems. The unit has three versions which differ in the way the problems are terrible, bait [version Y], contrived [version W], derp [version X].
A difficult unit consisting of problems with deceptively short statements. A lot of the difficulty of these problems is setting up an entire framework to attack a simply stated problem. These setups are often more elaborate or detailed than in other problems.
A short end-of-year unit containing open problems I solved at the Duluth REU.
The worst olympiad problems you've ever seen. The name is a reference to the video game term in which you do the same thing over and over.
Computational problems in modular arithmetic, again at the border of short-answer contests and olympiads.
A unit about the idea that if d divides a and b, then d divides any linear combination of a and b. This intuition underlies Bezout's lemma, the Euclidean algorithm, and the division algorithm, as well as a techinque which I privately call remainder bounding. One very good example of a problem of this feeling is SL 2016 N4 (sort of the crown example of remainder bounding).
Expressions of the form a^n \pm 1, the bread and butter of olympiad number theory. Mods and orders, Fermat's Christmas theorem, lifting the exponent.
Heavy machinery in number theory: Vieta jumping, quadratic reciprocity, and some big-name theorems you may or may not have heard of.
A theoretical unit on the algebra and number theory of polynomials over Z, bordering into some algebraic number theory and Galois theory. Algebraic integers and irreducible polynomials feature prominently in this unit. The number theory in this unit goes deeper than that used in the Irreducible unit.
Construction problems in number theory. In some ways it's like the free unit because you get to make some decisions, but in other ways Z has a lot of structure that you might know things about, and you'll have to balance these two intuitions.
Orders modulo a prime, at the border of AIME and USA(J)MO but leaning a lot more towards the latter. Intended as an introduction into olympiad number theory.
The use of v_p in handling olympiad problems.
Size in NT
Using size as a way to handle number theory conditions, for example taking sufficiently large primes. On the border between olympiad algebra and olympiad number theory.
Number theory practice for experts, combining problems from Exp and Heavy NT as well as some other sources.