This is the public-facing unit catalog and art gallery for OTIS. Last updated Tue, 17 Dec 2024 10:31:45 -0500.
Art contributed by Aaryan Vaishya.
Problems that purely involve algebraic manipulations without some other context, e.g. solving rather arbitrary equations or systems of equations.
Versions offered: BAY, DAY. Completed by 249 students.
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 42 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 237 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 35 students.
Art contributed by Filbert Wu.
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.
Versions offered: BAX. Completed by 140 students.
Art contributed by Isken Kenzhebaev.
A unit featuring some of the hardest gems from the golden age of inequalities.
Versions offered: ZAW, ZAY. Completed by 22 students.
Art contributed by Alan Cheng.
The basics of inequalities: AM-GM, homogenization, Cauchy/Holder.
Versions offered: BAW. Completed by 210 students.
Art contributed by Heyang Ni.
Techniques for inequalities of the form $f(x_1) + ... + f(x_n)$. Jensen, Karamata, tangent line trick. The $n-1$ equal value principle is deferred to Ineq Standard.
Versions offered: BAW. Completed by 92 students.
Art contributed by Shrrivathsa Mahesh.
Inequalities which can be approached using all the standard methods. This is sort of a combination of Ineq Basic and Ineq Func.
Versions offered: DAX. Completed by 124 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 33 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 41 students.
Art contributed by Owen Zhang.
An extension of the Sums unit --- rather than just swapping infinite sums, we now get to enjoy swapping infinite integrals as well.
Versions offered: ZAW. Completed by 13 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 79 students.
Art contributed by Jacopo Rizzo.
Practice with manipulating sums, and in particular switching the order of summation (or integration). Features generating functions and Snake Oil as well.
Versions offered: BAX, DAX. Completed by 181 students.
Art contributed by Rohan Dhillon.
Vieta formulas, Newton sums, and the fundamental theorem of symmetric polynomials. Involves some computational problems. In my opinion, this is probably the easiest unit.
Versions offered: BAW. Completed by 359 students.
Art contributed by Joel Gerlarch.
Harder inequality problems that don't succumb to the standard methods: it takes some more ingenuity to figure out how to approach these.
Versions offered: DAX. Completed by 47 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 27 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 77 students.
Art contributed by Owen Zhang.
A challenging combinatorics unit to conclude the year, with difficult problems reviewing everything that has appeared earlier.
Versions offered: ZCW. Completed by 13 students.
Art contributed by Evan Chen.
The "ARML combo" unit, this counting unit represents the centroid of Rigid, Induct, Formulas, and Grinding. Also contains several black-magic bijections.
Versions offered: BCW, DCW. Completed by 79 students.
Art contributed by Rohan Dhillon.
A beginner combinatorics unit, meant to help get people oriented with typical proof styles for olympiad problems. Induction, recursion, invariants, and algorithms.
Versions offered: BCX, BCY. Completed by 396 students.
Art contributed by Owen Zhang.
An important unit about taking advantage of the equality case in combinatorial problems in order to solve problems. Mandatory for newcomer students.
Versions offered: BCW, DCW, ZCW. Completed by 318 students.
Art contributed by Heyang Ni.
Computational problems involving probability, expected value (in particular linearity of expectation), and Markov chains (processes which move from state to state).
Versions offered: BCW. Completed by 270 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 39 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 98 students.
Art contributed by Anthony Zou.
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.
Versions offered: BCW, DCW, DCX. Completed by 447 students.
Art contributed by Azat Madimarov.
An accelerated version of both the global and local units (both done at once).
Versions offered: ZCW, ZCX. Completed by 82 students.
Art contributed by Owen Zhang.
Combinatorics practice with graphs. The B version of the unit is aimed at students with no previous exposure to graph theory; the harder versions assume prior experience.
Versions offered: BCX, DCX, ZCX. Completed by 176 students.
Art contributed by Cecilia Sun.
A fun but difficult unit on combinatorics problems involving rectangular grids.
Versions offered: ZCY. Completed by 26 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 137 students.
Art contributed by Soumitro Shovon Dwip.
This is a duplicate of Induction I, but more difficult and with different artwork.
Versions offered: DCY. Completed by 64 students.
Art contributed by DALL·E.
Mixed combinatorics practice at the IMO 2/5 level.
Versions offered: DCW, DCX. Completed by 30 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 42 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 53 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 234 students.
Art contributed by Alon Ragoler.
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.
Versions offered: BCW. Completed by 7 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 76 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 208 students.
Art contributed by Cecilia Sun.
A fun unit involving combinatorics problems from Russia.
Versions offered: BCX, DCX. Completed by 67 students.
Art contributed by Emily Yu.
Functional equations, I guess.
Versions offered: BFW, DFW, ZFW, DFX, ZFX. Completed by 506 students.
Art contributed by Owen Zhang.
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.
Versions offered: DFW, ZFW. Completed by 65 students.
Art contributed by Lum Jerliu.
On real-valued functional equations in which all variables are "wrapped" by the function in some way.
Versions offered: DFY. Completed by 64 students.
Art contributed by Evan Chen.
Computational geometry problems, many taken from the tail end of the AIME. At the border of computational contests and olympiads.
Versions offered: BGX, DGX. Completed by 205 students.
Art contributed by Evan Chen.
This is a unit about building "diagram intuition": being able to take a geometry diagram (which may be good or bad) and trying to get a sense of which claims should or shouldn't be true.
Versions offered: DGX, ZGY. Completed by 142 students.
Art contributed by Evan Chen.
Barycentric coordinates in olympiad geometry. A follow-up to Chapter 7 of EGMO.
Versions offered: BGW, DGW, DGX. Completed by 99 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 117 students.
Art contributed by Evan Chen.
Complex numbers in olympiad geometry. A follow-up to Chapter 6 of EGMO.
Versions offered: BGW, DGW, DGX. Completed by 184 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 109 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 437 students.
Art contributed by Aditya Pahuja.
A challenging geometry unit to conclude the year, with difficult problems reviewing everything that has appeared earlier.
Versions offered: ZGY. Completed by 16 students.
Art contributed by Nurtilek Duishobaev.
Your friendly projective geometry unit. Harmonic bundles, poles and polars, and so on. A follow-up to Chapter 9 of EGMO.
Versions offered: DGW, DGX. Completed by 405 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 116 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 35 students.
Art contributed by anonymous.
A silly (but difficult) unit on the theory of rectangular circumhyperbolas and the Poncelet point. For fun, if you really like hardcore projective geometry.
Versions offered: ZGX. Completed by 26 students.
Art contributed by Emily Yu.
A harder follow-up unit to chapters 8 and 10 of EGMO.
Versions offered: ZGY. Completed by 46 students.
Art contributed by Heyang Ni.
Inversion in olympiad geometry, following up chapter 8 of EGMO.
Versions offered: DGW. Completed by 90 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 Alan Cheng.
A technique involving animating points and considering resulting projective maps. This lecture was contributed by Anant Mudgal.
Versions offered: ZGW. Completed by 26 students.
Art contributed by Lum Jerliu.
Spiral similarity and Miquel points, following up chapter 10 of EGMO.
Versions offered: DGW. Completed by 75 students.
Art contributed by Evan Chen.
A more difficult version of the barycentric coordinates unit.
Versions offered: ZGX. Completed by 17 students.
Art contributed by Evan Chen.
A more difficult version of the complex numbers unit.
Versions offered: ZGX. Completed by 19 students.
Art contributed by DALL·E.
The trig-bash unit. Several traditional-style geometry problems that are meant to be solved by chasing lengths and ratios, with the aid of trigonometric techniques.
Versions offered: DGW. Completed by 19 students.
Art contributed by Emily Yu.
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.
Versions offered: DGX, ZGX. Completed by 29 students.
Art contributed by DALL·E.
A first introduction to the language of group theory, rings, and fields. One of the two key gateway units for higher math.
Versions offered: DHW. Completed by 2 students.
Art contributed by DALL·E.
A quick introduction to important notions like continuity, compactness, topological spaces, etc. One of two key gateway units for higher math.
Versions offered: DHW. Completed by 1 student.
Art contributed by Aaryan Vaishya.
A unit consisting entirely of troll "anti-problems" which are suitable for giving to your enemies.
Versions offered: DMW, DMX, DMY. Completed by 57 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 53 students.
Art contributed by anonymous.
A short end-of-year unit containing open problems I solved at the Duluth REU.
Versions offered: ZMY. Completed by 2 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 423 students.
Art contributed by Anthony Zou.
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.
Versions offered: DMY, ZMY. Completed by 21 students.
Art contributed by Sambhu Ganesan.
Computational problems in modular arithmetic, again at the border of short-answer contests and olympiads.
Versions offered: BNW. Completed by 291 students.
Art contributed by Nurtilek Duishobaev.
Problems that involve asymptotic calculations in number theory, featuring some multiplicative number theory. Convolution method, and generally problems that require more technical estimates.
Versions offered: ZNW. Completed by 6 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 128 students.
Art contributed by Owen Zhang.
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.
Versions offered: DNW. Completed by 179 students.
Art contributed by Rohan Garg.
Heavy machinery in number theory: Vieta jumping, quadratic reciprocity, and some big-name theorems you may or may not have heard of.
Versions offered: DNW. Completed by 111 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 52 students.
Art contributed by Emily Yu.
Miscellaneous number theory problems that didn't fit well in other units.
Versions offered: DNX. Completed by 47 students.
Art contributed by ``Milk''.
A challenging number theory unit to conclude the year, with difficult problems reviewing everything that has appeared earlier.
Versions offered: ZNY. Completed by 13 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 86 students.
Art contributed by Rohan Garg.
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.
Versions offered: BNW. Completed by 280 students.
Art contributed by Heyang Ni.
The use of $\nu_p$ in handling olympiad problems.
Versions offered: BNW, DNW. Completed by 288 students.
Art contributed by Anurag Singh.
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.
Versions offered: DNW, DNX. Completed by 117 students.
Art contributed by Sambhu Ganesan.
Number theory practice for experts, combining problems from Exp and Heavy NT as well as some other sources.
Versions offered: ZNW, ZNX. Completed by 55 students.