This is the public-facing unit catalog for OTIS. Last updated Thu, 17 Oct 2024 12:18:15 -0400.

Art contributed by Aditya Pahuja.

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.

Versions offered: DAY, ZAY. Completed by 39 students.

Art contributed by Aaryan Vaishya.

Roots of unity. Using $e^{i \theta} = \cos(\theta) + i \sin(\theta)$, and connecting them to trigonometry, etc. Features a ton of problems from Math Prize for Girls. Despite the name of the unit, cyclotomic polynomials themselves don't appear very often; the idea of roots of unity is much more prevalent.

Versions offered: BAY. Completed by 228 students.

Art contributed by Alex Zhao.

Based on Yang Liu's class "Write Down Formulas" at MOP 2018. This unit consists of problems which involve manipulations of fairly involved formulas, such as combinatorial recursions or number theoretic power sums. Despite officially being algebra, there are just as many (maybe more) problems that would be classified as C or N.

Versions offered: DAW, ZAW. Completed by 31 students.

Art contributed by Evan Chen.

Was your calculus class too easy for you? Do you want to stare at random artificial expressions having no idea how to find their antiderivative? If so, this unit is for you! Enjoy a student-contributed guest unit featuring problems from integration bees from MIT and other places.

Versions offered: DAX. Completed by 31 students.

Art contributed by Azat Madimarov.

Problems about showing polynomials are irreducible; these are rare in olympiads, but give you a lot of good intuition about how $\mathbb{Z}[x]$ polynomials behave. Techniques that appear include working in $\mathbb{F}_p[x]$, looking at the magnitude of complex roots a la Rouche, and other ad-hoc tricks. More olympiad algebra than the integer polynomials unit.

Versions offered: DAX. Completed by 40 students.

Art contributed by Owen Zhang.

General polynomials unit, maintaining some distance from integer polynomials (though still overlapping slightly). Includes Vieta/Newton, multivariable polynomials, Lagrange interpolation, size arguments, differentiation.

Versions offered: DAW, ZAW. Completed by 70 students.

Art contributed by Arul Kolla.

Formerly, this was a unit on using combinatorial nullstellensatz, mostly for fun. It was later expanded to additionally include uses of generating functions and related polynomial methods on 3/6 level problems, broadening the scope significantly.

Versions offered: ZCW. Completed by 26 students.

Art contributed by Alon Ragoler.

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.

Versions offered: DCW, ZCW. Completed by 73 students.

Art contributed by Joel Gerlach.

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.

Versions offered: DCX, ZCX. Completed by 38 students.

Art contributed by Arul Kolla.

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).

Versions offered: DCW, ZCX. Completed by 92 students.

Art contributed by Jiya Dani.

Problems which really use induction and recursion in 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.

Versions offered: BCX. Completed by 125 students.

Art contributed by Soumitro Shovon Dwip.

Algorithmic problems which involve showing that it is possible to achieve some task (rather than finding invariants or proving impossibility). Features selected problems from the IOI, so some CS background is helpful but not necessary.

Versions offered: DCY, ZCY. Completed by 41 students.

Art contributed by Lum Jerliu.

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 $\mathbb R$ and linear algebra over $\mathbb{F}_2$.

Versions offered: DCY, ZCY. Completed by 50 students.

Art contributed by Emily Yu.

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.

Versions offered: DCW, DCX. Completed by 218 students.

Art contributed by Owen Zhang.

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.

Versions offered: DCY, ZCY. Completed by 73 students.

Art contributed by Arul Kolla.

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).

Versions offered: BCX, DCX, ZCX. Completed by 198 students.

Art contributed by Evan Chen.

Formerly part of "American Geo". This is somewhere between Config Geo and Elem Geo. 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 certain picture.

Versions offered: DGX, ZGX. Completed by 111 students.

Art contributed by Evan Chen.

Formerly known as "American Geo". This is 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 were popular on the USA olympiads and team selection tests around 2016 and it is totally not my fault. These particular ones tend to use common or standard configurations as a base and build on top of them, as opposed to starting afresh.

Versions offered: DGY, ZGY. Completed by 99 students.

Art contributed by Rishabh Mahale.

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.

Versions offered: BGW, DGW, BGY, DGY. Completed by 397 students.

Art contributed by Nurtilek Duishobaev.

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.

Versions offered: DGY, ZGY. Completed by 108 students.

Art contributed by Joel Gerlach.

Grab-bag of geometry problems that feature some algebra, combinatorics, or number theory. As examples, this includes geometric inequalities, combinatorial geometry, and problems involving integer distances or lattice points.

Versions offered: DGX, ZGX. Completed by 34 students.

Art contributed by Gunjan Aggarwal.

This unit revolves around two particular techniques: linearity of a difference of power of a point and the forgotten coaxiality lemma. This is used to compare powers of a point with respect to two different circles, particularly showing they are equal.

Versions offered: ZGY. Completed by 8 students.

Art contributed by Emily Yu.

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.

Versions offered: DMW, ZMW. Completed by 50 students.

Art contributed by Evan Chen.

Not an actual unit; used internally as a canonical file for testing. If you unlock this unit, you will get a blank file. NOTHING TO SEE HERE MOVE ALONG.

More seriously you can use this to test the submission interface too if you are new to OTIS.

Versions offered: BMW. Completed by 409 students.

Art contributed by Alex Zhao.

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 technique 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).

Versions offered: DNY. Completed by 120 students.

Art contributed by Aaryan Vaishya.

A theoretical unit on the algebra and number theory of polynomials over $\mathbb Z$, 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.

Versions offered: DNY, ZNY. Completed by 49 students.

Art contributed by Heyang Ni.

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.

Versions offered: DNY, ZNY. Completed by 84 students.