The OTIS Excerpts - Evan Chen

1 The otis ExcerptsA collection of 192 problems and solutionsEvan ChenJanuary 23, noblest art is that of making others T. BarnumIf you like this book and want to support me,please consider buying me a coffee! 2019 Evan Chen. All rights reserved. Personal use book is a selection of notes from twelve of the lectures that I use inmy year-round math olympiad classes. The program is affectionately namedOTIS. The abbreviation officially stands for Olympiad Training for IndividualStudy ; but in truth, I rigged the acronym so that it would match the nameof an elevator in Lincoln, Nebraska of which I had fond childhood I started teaching otis back in the fall of 2015, it was just a smallgroup of students at Phillips Academy that I would meet with every couple ofweeks. Despite my inexperience at the time, I have fond memories of this firstgroup, and I maybe learned as much from them as they did from year since then, the number of otis students has doubled, and thenumber of applications has increased even faster than that.

2 At the same time,I became increasingly involved with volunteering with the organization of high-school contests. By the time I started graduate school, I was spending so littletime on my own studies that I began to fear (rightly or wrongly) that I mightfail out of the math PhD , despite my best efforts, I eventually had the heartbreaking task ofhaving to tell eager and enthusiastic students that I simply did not have thetime or space to take them all under my wing. I am the kind of person whofinds it hard to say no, and so this was painful for me to do. otis taught methe reality that I am just one this way, these notes are an apology to everyone I turned down, and toeveryone that I will have to turn down. I would have loved to be able to helpeveryone who came to my doorstep.

3 I am sorry that I could not do more, butI wrote you a short book, as it was the least I could with all my works, there are bound to be numerous errors, mistakes,or points of unclarity. Comments, suggestions, corrections are very can reach me ChenDecember 30, 2018 Fremont, California, USAiiiIntroductionThe book is divided into algebra, combinatorics , and number theory. We donot cover geometry, for whichEuclidean Geometry in Mathematical Olympiads[Che16] already serves the role of comprehensive book .The twelve main chapters in this book are structured in to four sections. Atheoretical portion, of varying length, in which relevant theoremsor ideas are developed. Some of this material is new, but the majority ofit is not. Most of it has been adapted, edited, and abridged from existinghandouts that you can still find general, the theoretical material here tries to stick to the basics, ratherthan being comprehensive.

4 A couplewalkthroughs. These are olympiad problems which are chosento illustrate ideas, accompanied by an outline of the designing my lecture notes for otis , I wrote these walkthroughswith the idea of emulating a person. In a real classroom the student doesnot simply passively listen to solutions. The process is more interactive:the instructor walks a student through the example, but with a back-and-forth series of prepared questions. My hope with the walkthroughsis to simulate this as best I can with static text. A series ofproblems. These problems cover a range of difficulties. Butin general, the first half of the problems in each chapter are intendedto be fairly accessible, perhaps at the level of IMO 1/4. The difficultyincreases quickly after that, with the closing problem usually being quitechallenging.

5 Fullsolutionsto both the walkthroughs and problems. (Great for in-flating page count!) Readers are encouraged to read solutions even toproblems that they solved; comments, remarks, or alternate solutionsfrequently addition, at the end of each part, a handful of problems chosen from USAselection tests are given, mostly for general, I assume the reader has some minimal experience with readingand writing proofs. However, I nonetheless dedicated the first chapter to somemathematical and stylistic comments which may be helpful to beginners inproofs. Readers with significant proof experience should feel no shame inskipping this first 23, 2022 The otis Excerpts , by Evan ChenContest abbreviationsMany problems have a source quoted, but there are a large number of abbre-viations as a result.

6 We tabulate some of the abbreviations Invitational Math Exam, the qualifying exam for the USAnational Girl s Math Olympiad (not to be confused with [Che16])ELMOThe ELMO is a contest held at the USA olympiad training camp everyyear, written by returning students for meaning of the acronym changes each year. It originally meant Experimental Lincoln Math Olympiad but future names have included elogMath Olympiad , End Letter Missing , Ex-Lincoln Math Olympiad , English Language Master s Open . Ego Loss May Occur , vErybadLy naMed cOntest , Eyyy LMaO .ELMO ShortlistA list of problems from which each year s ELMO is Math Tournament, the largest collegiate math compe-tition in the United States. The contest is held twice a year, in Novemberand Math Olympiad, the supreme high-school ShortlistA list of about 30 problems prepared annually, from which thesix problems of the IMO are selected by William Lowell Putnam Mathematical Competition, an annualcompetition for undergraduate students studying in USA and Masters in Mathematics, an annual olympiad held in Ro-mania in late February for teams with a strong performance at the In-ternational Mathematical embarrassingly named Team Selection Test Selection Test.

7 Held in June each year, the TSTST selects students for the USA TeamSelection for Team Selection Test. Most countries use a TST asthe final step in the selection of their team for the International Junior Math Olympiad, the junior version of the nationalmath olympiad for the United States (for students in 10th grade andbelow).USAMOUSA Math Olympiad, the national math olympiad for the Notes on Common proof mistakes .. Find all problem are always two-part problems .. for reversibility .. problems are always two-part problems .. neat, be careful .. Stylistic writing suggestions .. on the level of detail .. write wrong math .. the point where you cross the ocean .. space .. Problems .. Solutions.

8 14 IAlgebra172 Fundamentals of Brief warning for beginners .. flipped inequalities .. chains of inequalities .. Polynomial inequalities .. and Muirhead .. vague cheerleading .. s inequality .. inequalities .. Three polynomial tricks .. special case of product 1 .. substitution .. s inequality .. Eliminating radicals and fractions .. Power Mean .. and H older .. Inequalities in arbitrary functions .. and Karamata .. line trick .. Walkthroughs ..281 January 23, 2022 The otis Excerpts , by Evan Problems .. Solutions ..323 Functional Definitions .. the generality of functions .. types of functions .. First example .. Second example (or non-example) .. Four techniques for motivating substitutions.

9 Cancellation .. fff trick .. parts .. Cauchy s equation .. s equation overQ.. s equation overR.. Walkthroughs .. Problems .. Solutions ..544 Monstrous Functional introduction .. Clues .. Linear algebra terminology .. Cauchy s equation overR.. to earth .. Walkthroughs .. Problems .. Solutions ..725 Selected Algebra from USA Problems .. Solutions ..84II Combinatorics956 Graph Theory Textbook references .. Graphs .. Degree .. Paths, cycles, connectedness ..992 ContentsJanuary 23, 20227 A simple example, the handshake lemma .. Expected value .. and notation .. motivating example .. of expectation .. The so-called pigeonhole principle.

10 Walkthroughs .. Problems .. Solutions .. 1108 Synopsis .. Walkthroughs .. Problems .. Solutions .. 1249 Synopsis .. Walkthroughs .. Problems .. Solutions .. 14210 Synopsis .. Walkthroughs .. Problems .. Solutions .. 15611 Walkthroughs .. Problems .. Solutions .. 16812 Selected combinatorics from USA Problems .. Solutions .. 176 III Number Theory18713 Definition and examples of order .. Application: Fermat s Christmas theorem .. Primitive roots .. 1903 January 23, 2022 The otis Excerpts , by Evan Walkthroughs .. Problems .. Solutions .. 19414 Look at the Definition.