# OTIS Catalog

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. The first letter of the code represents the difficulty, the second letter repeats the olympiad category, and the third letter is a version identifier. Many topics have multiple versions so they can be repeated in different years.

## Algebra

Analysis [Algebra, DAY ZAY].
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.

Gen Func [Algebra, BAX].
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.

Hard Ineq [Algebra, ZAW ZAY].
A unit featuring some of the hardest gems from the golden age of inequalities.

Ineq Basic [Algebra, BAW].
The basics of inequalities: AM-GM, homogenization, Cauchy/Holder.

Ineq Func [Algebra, BAW].
Techinques for inequalities of the form f(x_1) + ... + f(x_n). Jensen, Karamata, tangent line trick, and n-1 equal value principle.

Ineq Standard [Algebra, DAX].
Inequalities which can be approached using all the standard methods. This is sort of a combination of Ineq Basic and Ineq Func.

Irreducible [Algebra, DAX].
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.

Sums [Algebra, BAX DAX].
Practice with manipulating sums, and in particular switching the order of summation (or integration). Features generating functions and Snake Oil as well.

Symm Polynom [Algebra, BAW].
Vieta formulas, Newton sums, and the fundamental theorem of symmetric polynomials. Involves some computational problems.

Tricky Ineq [Algebra, DAX].
Harder inequality problems that don't succumb to the standard methods: it takes some more ingenuity to figure out how to approach these.

## Combinatorics

Arrows [Combinatorics, DCW ZCW].
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.

Combo Null [Combinatorics, ZCW].
A silly unit on using combinatorial nullstellensatz. Mostly for fun.

Equality [Combinatorics, DCW ZCW].
An important unit about taking advantage of the equality case in combinatorial problems in order to solve problems. Mandatory for newcomer students.

Expected Value [Combinatorics, BCW].
Computational problems involving probability, expected value (in particular linearity of expectation), and Markov chains (processes which move from state to state).

Extremal Graph [Combinatorics, ZCX].
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 and I think this is the most difficult combinatorics unit in all of OTIS.

Free [Combinatorics, 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).

Global [Combinatorics, 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 [Combinatorics, ZCW ZCX].
An accelerated version of both the global and local units (both done at once).

Graph Theory [Combinatorics, DCX].
Combinatorics practice with graphs.

Hall [Combinatorics, BCW DCW].
Hall's marriage lemma.

Induct [Combinatorics, 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.

Linear Algebra [Combinatorics, DCY ZCY].
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.

Local [Combinatorics, DCW DCX].
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.

Mystery [Combinatorics, BCW].
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.

Process [Combinatorics, DCY ZCY].
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.

Rectangular Grids [Combinatorics, ZCY].
A fun but difficult unit on combinatorics problems involving rectangular grids.

Rigid [Combinatorics, 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).

Russian Combo [Combinatorics, DCX].
A fun unit involving combinatorics problems from Russia.

## Functional Equations

Func Eqn [Functional Equations, BFW DFW DFX ZFX ZFY].
Functional equations, I guess.

Monster FE [Functional Equations, DFW ZFW].
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.

## Geometry

AIME Geo [Geometry, BGY BGX].
Computational geometry problems, many taken from the tail end of the AIME. At the border of computational contests and olympiads.

American Geo [Geometry, DGX ZGX ZGY DGY].
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.

Bary [Geometry, BGW DGW DGX].
Barycentric coordinates in olympiad geometry. A follow-up to Chapter 7 of EGMO.

Complex Nums [Geometry, BGW DGW DGX].
Complex numbers in olympiad geometry. A follow-up to Chapter 6 of EGMO.

Elementary Geo [Geometry, BGW 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.

Harmonic [Geometry, DGW DGX ZGW].
Your friendly projective geometry unit. Harmonic bundles, poles and polars, and so on. A follow-up to Chapter 9 of EGMO.

Homography [Geometry, ZGY].
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.

Hyperbola [Geometry, ZGX].
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 [Geometry, ZGW ZGY].
A harder follow-up unit to chapters 8 and 10 of EGMO.

Invert [Geometry, DGW].
Inversion in olympiad geometry, following up chapter 8 of EGMO.

Spiral [Geometry, DGW].
Spiral similarity and Miquel points, following up chapter 10 of EGMO.

Super Bary [Geometry, ZGX].
A more difficult version of the barycentric coordinates unit.

Super Complex [Geometry, ZGX].
A more difficult version of the complex numbers unit.

Weird Geo [Geometry, DGX ZGX].
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.

## Miscellaneous

Anti Problems [Miscellaneous, DMW DMX].
A unit consisting entirely of troll hit/miss problems.

Courage [Miscellaneous, DMW ZMW].
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.

Duluth [Miscellaneous, ZMY].
A short end-of-year unit containing open problems I solved at the Duluth REU.

Grinding [Miscellaneous, ZMY DMY].
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.

## Number Theory

AIME Mods [Number Theory, BNW].
Computational problems in modular arithmetic, again at the border of short-answer contests and olympiads.

Diophantine [Number Theory, DNW].
Nominally a unit about Diophantine equations, but with a focus on expressions of the form a^n \pm 1. Mods and orders, Fermat's Christmas theorem, lifting the exponent.

Euclid Algorithm [Number Theory, BNY DNY].
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).

Heavy NT [Number Theory, DNW].
Heavy machinery in number theory: Vieta jumping, quadratic reciprocity, and some big-name theorems you may or may not have heard of.

Int Polynom [Number Theory, DNY ZNY].
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.

NT Construct [Number Theory, DNY ZNY].
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 [Number Theory, BNW].
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.

Prime Exponents [Number Theory, BNW].
The use of v_p in handling olympiad problems.

Size in NT [Number Theory, DNW].
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.

Super NT [Number Theory, ZNW ZNX].
Number theory practice for experts, combining problems from Dioph and Heavy NT as well as some other sources.