Suggestions for starting math olympiads

来源: ca981 于 2018-02-02 12:40:39 [] [博客] [旧帖] [给我悄悄话] 本文已被阅读： 0 次 (155614 bytes)
回答: 弹琴与中文 由 ca981 于 2018-02-01 06:22:19

 

http://web.evanchen.cc/recommend.html

Reading Recommendations

The world would be a great place if I could write about everything I knew about, but alas I have a finite amount of time. So in addition to the stuff I have on this website, here's a list of other resources I like.

Please notify me of any broken links, suggestions, etc. by email.

An abridged version of this page for olympiad students can be found here.

Undergraduate Math and Computer Science

  • MSci Category Theory notes by Tom Leinster. I highly enjoyed these notes; very carefully written and explains intuition. Some minimal knowledge of group theory and linear algebra is used in the examples. See instead the corresponding print book.

  • Analytic NT notes by AJ Hildebrand. A set of lecture notes for analytic number theory, suitable for self-study. A light introduction where you get to prove versions of the Prime Number Theorem and Dirichlet's Theorem.

  • Algebraic Geometry by Andreas Gathmann. My preferred introduction to algebraic geometry; short but complete. This was the source that finally got me to understand the concept of a ringed space. It doesn't officially cover schemes, but because it covers general varieties as ringed spaces the full-fledged scheme is not much harder.

  • Manifolds and Differential Forms by Reyer Sjamaar. My preferred introduction to differential geometry; very readable and works with minimal prerequisites. Also, beautifully drawn figures.

  • Harvard's CS 125: Algorithms and Complexity has delightful lecture and section notes.

Of course, see also Napkin.

Olympiad Resources

Handouts

  • My own handouts (sorry, couldn't resist linking them again).

  • Yufei Zhao's site has several excellent handouts, especially in geometry. I consulted many of them when I was coming up with ideas for my geometry textbook. In particular, the Cyclic Quadrilaterals handout is especially worth reading.

  • Alexander Remorov, in particular the projective geometry handout, which the corresponding chapter in my textbook is based off of.

  • Po-Shen Loh, mostly combinatorics. See especially the handouts on the probabilistic method.

Books

Contests

Each section is in alphabetical order.

 

http://web.evanchen.cc/wherestart.html

Suggestions for starting math olympiads

By now I've gotten countless emails from students studying for math olympiads that amounts to "okay, but what do I do?". So here is, at long last, a list of suggestions.

Readers may notice this is actually just an abridged version of the longer recommendations page on my site. It turns out if you make a list shorter, people are more likely to pick up the pencil and start cracking.

I really want to stress these are mere suggestions. Just because you have done everything on this list does not mean you will achieve your goals. Conversely, there are many fantastic resources that are not included on this list, since I wanted to keep this list very short (and also due to my own ignorance). If you are looking for a list of materials which are guaranteed to be "enough" for solving IMO #1 and #4, you have come to the wrong place. There are few guarantees in math contests.

See also the math contest FAQ's for some more philosophical (and less concrete) advice on studying.

Lecture Materials

If you've finished these and want more, I have a bunch more olympiad handouts and links to other sources.

Problem Sources

A mandatory part of any olympiad training is doing lots of problems from past contests. (I used to carry a binder with printouts of the IMO shortlist and check them off as I figured them out.)

Here are some places to start (roughly ascending order of difficulty):

The bottom of the recommendations page has some more suggestions for problems if this list isn't sufficient.

 

 

 

My own handouts (sorry, couldn't resist linking them again).

http://web.evanchen.cc/olympiad.html

Olympiad Articles

Over time I've written a few handouts pertaining to olympiad math. Here is a list of them. A few are written in traditional Chinese as I was practicing my Chinese in preparation for the Taiwan TST.

The Math and Problem Solving sections of my personal blog might also be of interest. See also Recommendations for other authors I like, as well as my geometry book for a comprehensive textbook in Euclidean geometry. See also Problems for contest papers.

To compile these documents in LaTeX, you will need evan.sty. Because this style file evolves over time, your output might look a little different than the PDF's attached here. For files with Asymptote diagrams you will additionally need olympiad.asy and cse5.asy.

If you notice any errors, please let me know!

General

  • English (pdf) (tex)
    Notes on proof-writing style. These were used at MOP 2016.

  • USAMO 2014 Contest Analysis (pdf) (tex)
    This describes in detail the thought process behind each of my solutions to the USAMO 2014.Click here for a copy of the solutions I submitted.

  • IMO 2014 Journal (pdf) (tex)
    This describes my experiences competing as TWN2 at the 55th IMO 2014.
    To download the pictures in the report, locate "media" in the source folder.

  • Taiwan TST 2014 Reflection (pdf) (tex)
    This describes my experiences competing for a position on the Taiwan IMO 2014 team. It also contains an extensive commentary on each of the Team Selection Tests and Quizzes, which together covered most of the 2013 IMO Shortlist. 
    Note: Click here for English versions of the TST problems.

Algebra and Number Theory

  • Introduction to Functional Equations (pdf) (tex)
    An introduction to functional equations for olympiad students.

  • Monsters (pdf) (tex)
    A handout discussing pathological functional equations.

  • Summation (pdf) (tex)
    General discussion of sums and products, and how to deal with them. Includes generating functions.

  • The Chinese Remainder Theorem (pdf) (tex)
    An article on the Chinese Remainder "Theorem".

  • Orders Modulo a Prime (pdf) (tex)
    Article on orders and primitive roots, in particular featuring the sum of squares lemma and its generalization to arbitrary cyclotomic polynomials.

Combinatorics

Geometry

  • Barycentric Coordinates in Olympiad Geometry (full) (abridged) 
    One of my most famous handouts. Introduces from scratch the method of barycentric coordinates. A cleaned-up version of it is found in the draft of my geometry textbook.

  • How to Use Directed Angles (pdf) (tex)
    A short note on the use of directed angles in olympiad solutions. Why didn't anyone beat me to writing this?

  • The Incenter/Excenter Lemma (pdf) (tex)
    A collection of problems which exhibit the first olympiad configuration I got to know well, the famous "incenter/excenter lemma".

  • Bashing Geometry with Complex Numbers (pdf) (tex)
    English translation of my original notes in Chinese. Describes the classic method of complex numbers.
    Original Chinese version: (pdf) (tex)

  • A Guessing Game: Mixtilinear Incircles (pdf) (tex)
    A quick description of some nice properties of mixtilinear incircles. Presented as a "guessing game" where one has to guess collinear points, cyclic quadrilaterals, and so on beforehand.

  • Writing Olympiad Geometry Problems (pdf) (tex)
    For students who are interested in writing their own olympiad geometry problems! Or more generally, anyone who is curious how my geometry problems get created.

Also: chapter 2 (on power of point) or chapter 8 (on inversion) of my textbook.

Inequalities

  • Lagrange Multipliers Done Correctly (pdf) (tex)
    This is a description of the conditions necessary to execute a Lagrange Multipliers solution on an olympiad.

  • SOS: A Dumbass's Perspective (pdf) (tex)
    Describes the SOS method for solving inequalities.

  • Olympiad Inequalities (pdf) (tex)
    English translation of my original notes in Chinese. Describes some "standard strategies" for handling olympiad inequalities.
    Original Chinese version: (pdf) (tex)

Miscellaneous

  • OMO Spring 2014 Executive Report (pdf) (tex)
    A short report on running the Online Math Open for Spring 2014. This is mostly for the sake of those who will be running the future installments of the contest, as one of us is graduating.

  • Chinese Terminology Sheet (pdf) (tex)
    These are the notes that I took when I was studying traditional Chinese in preparation for the Taiwan IMO selection and training camps. It is updated occasionally as I add entries (e.g. when translating HMMT contests).

  • LaTeX Example File (pdf) (tex)
    A very short document which shows what a "typical" LaTeX file looks like. It was written for a complete beginner.

  • USAMO 2003 Rubric (pdf) 
    The grading rubric for USAMO 2003.

 

http://web.evanchen.cc/otis.html

Olympiad Training for Individual Study (OTIS)

Since 2015, I have run a small, informal training program during the school year for students aiming to do well on the USA(J)MO. It is now informally called OTIS. This program is designed for students who are comfortable reading and writing proofs and able to confidently qualify for national olympiad, e.g. consistently 10+ on AIME for American students. (Students at other levels should feel welcome to contact me for referrals to other mentors.)

See here for my thoughts on teaching philosophy. This page is a bit about the specifics of my program.

Description

OTIS is centered around two-week topic units which I pick for each student based on their background. Each semester has 6-7 units. Each unit comes with a problem set of 7-12 olympiad problems. Additionally, 10-14 olympiad-style practice exams (each emulating a day of a typical olympiad) are assigned throughout the year (graded in full).

All the materials for OTIS are hand-designed by me: the design of the materials is kind of an art, and it's half the fun of teaching (the other half is talking to the kids).

Typically, I meet with students over Google Hangouts either every two weeks, for about 75 minutes. I use LaTeX-Beamer as a blackboard and so full transcripts are posted immediately after each session. These sessions take place on weekdays, usually in the evening though international students may meet at other times.

The cost for each semester is 80(H+5) where H is the number of hours spent in person (the +5 term accounting for grading/preparation time). A rough estimate of the time commitment might be 8-12 hours per week, although there is a large variance.

Syllabus and Documents

For interested students

Here is the link to request form, due May 2, 2018. I select a handful of students from the many requests in mid-May of each year, after the grading of USAMO.

You can now submit requests for the 2018 - 2019 school year (to begin in September 2018). For OTIS 2018-2019, please submit requests by May 2, 2018. The form asks for the following information:

  • Your name, grade, school, and time zone
  • Your anticipated availabilities on each of Monday, Tuesday, Wednesday, Thursday
  • Any relevant contest history or scores
  • Subject preferences, like "strong geo, weak algebra"
  • Goals for the year, like "qualify for MOP"

The form will let you edit your responses after submission. In particular, you can submit the form early and then edit it after the USA(J)MO in April with your estimated scores.

Usually I am not able to take all (or even most) requests, but even then I can often refer you to other instructors who are less busy than me (mostly MOP alumni now at Harvard or MIT). So, if you are interested in mentoring, feel free to reach out to me even if the deadline is long past; most likely, I will be able to connect you with someone else.

History

I started teaching in 2015 when a group of parents from Phillips Andover Academy emailed me in early April, asking if I'd be interested in coaching a group of five of their students. At the time I called it "Andover Olympiad Training", and would make a trip up north from MIT every Sunday to work with them.

Since then word has gotten around, and more and more requests have come to me. I now teach 10-15 students each semester from both coasts, meeting them online during my evenings and the weekends, in lieu of studying for my actual classes. Thus my part-time job is to work with some of the most talented and motivated math students in the country on the same problems that I loved so much back in high school. I have the best job in the world.

Here are some awards that my past students have won. This is not a claim that these students performed well because of me; all of them were already strong before joining OTIS, and I think they would likely have done well even without me.

But sensei is still super proud of his kids even though he didn't do that much.

 

http://web.evanchen.cc/

Homepage of Evan Chen

About

I am an undergraduate at the Massachusetts Institute of Technology, graduating in June 2018. My CV is here. Here is a list of my major coursework.

Most of my research is either in number theory or enumerative/algebraic combinatorics. My papers are linked on my publications page.

I am actively involved with math olympiad competitions and have authored several olympiad materials, most notably including a geometry textbook. I also teach my own classes.

I am on GitHub and WordPress.

 

 

http://web.evanchen.cc/napkin.html

The Napkin Project

For most recent draft, click here to download.

Project Status

See link above for the most recent draft, and here for an archive of all drafts. I would very highly appreciate any corrections, suggestions, or comments.

Most of what remains to do is fill in the rest of the exercise problems, and slowly let all the typos get sent in to me.

(There is also a live Dropbox link which updates in real-time as I make changes. Feel free to email me to acquire this link.)

Description

The Napkin project is a personal exposition project of mine aimed at making higher math accessible to high school students. The philosophy is stated in the preamble:

I'll be eating a quick lunch with some friends of mine who are still in high school. They'll ask me what I've been up to the last few weeks, and I'll tell them that I've been learning category theory. They'll ask me what category theory is about. I tell them it's about abstracting things by looking at just the structure-preserving morphisms between them, rather than the objects themselves. I'll try to give them the standard example Gp, but then I'll realize that they don't know what a homomorphism is. So then I'll start trying to explain what a homomorphism is, but then I'll remember that they haven't learned what a group is. So then I'll start trying to explain what a group is, but by the time I finish writing the group axioms on my napkin, they've already forgotten why I was talking about groups in the first place. And then it's 1PM, people need to go places, and I can't help but think:

Man, if I had forty hours instead of forty minutes, I bet I could actually have explained this all.

This book is my attempt at those forty hours.

This project has evolved to more than just forty hours.

Current Table of Contents.

I. Basic Algebra and Topology

  • Chapter 1. What is a Group?
  • Chapter 2. What is a Space?
  • Chapter 3. Homomorphisms and Quotient Groups
  • Chapter 4. Topological Notions
  • Chapter 5. Compactness

II. Linear Algebra and Multivariable Calculus

  • Chapter 6. What is a Vector Space?
  • Chapter 7. Trace and Determinant
  • Chapter 8. Spectral Theory
  • Chapter 9. Inner Product Spaces

III. Groups, Rings, and More

  • Chapter 10. Group Actions Overkill AIME Problems
  • Chapter 11. Find All Groups
  • Chapter 12. Rings and Ideals
  • Chapter 13. The PID Structure Theorem

IV. Complex Analysis

  • Chapter 14. Holomorphic Functions
  • Chapter 15. Meromorphic Functions
  • Chapter 16. Holomorphic Square Roots and Logarithms

V. Quantum Algorithms

  • Chapter 17. Quantum states and measurements
  • Chapter 18. Quantum circuits
  • Chapter 19. Shor's algorithm

VI. Algebraic Topology I: Homotopy

  • Chapter 20. Some Topological Constructions
  • Chapter 21. Fundamental Groups
  • Chapter 22. Covering Projections

VII. Category Theory

  • Chapter 23. Objects and Morphisms
  • Chapter 24. Functors and Natural Transformations
  • Chapter 25. Abelian Categories

VIII. Differential Geometry

  • Chapter 26. Multivariable Calculus Done Correctly
  • Chapter 27. Differential Forms
  • Chapter 28. Integrating Differential Forms
  • Chapter 29. A Bit of Manifolds

IX. Algebraic Topology II: Homology

  • Chapter 30. Singular Homology
  • Chapter 31. The Long Exact Sequence
  • Chapter 32. More on Computing Homology Groups
  • Chapter 33. Bonus: Cellular Homology
  • Chapter 34. Singular Cohomology
  • Chapter 35. Applications of Cohomology

X. Algebraic NT I: Rings of Integers

  • Chapter 36. Algebraic Integers
  • Chapter 37. Unique Factorization (Finally!)
  • Chapter 38. Minkowski Bound and Class Groups
  • Chapter 39. More Properties of the Discriminant
  • Chapter 40. Bonus: Let's Solve Pell's Equation!

XI. Algebraic NT II: Galois and Ramification Theory

  • Chapter 41. Things Galois
  • Chapter 42. Finite Fields
  • Chapter 43. Ramification Theory
  • Chapter 44. The Frobenius Endomorphism
  • Chapter 45. A Bit on Artin Reciprocity

XII. Representation Theory

  • Chapter 46. Representations of Algebras
  • Chapter 47. Semisimple Algebras
  • Chapter 48. Characters
  • Chapter 49. Some Applications

XIII. Algebraic Geometry I: Varieties

  • Chapter 50. Affine Varieties
  • Chapter 51. Affine Varieties as Ringed Spaces
  • Chapter 52. Projective Varieties
  • Chapter 53. Bonus: Bezout's Theorem

XIV. Algebraic Geometry II: Schemes

  • Chapter 54. Morphisms of varieties
  • Chapter 55. Sheaves and Ringed Spaces
  • Chapter 56. Schemes

XV. Set Theory I: ZFC, Ordinals, and Cardinals

  • Chapter 57. Bonus: Zorn's Lemma and Cauchy's Functional Equation
  • Chapter 58. Zermelo-Frankel with Choice
  • Chapter 59. Ordinals
  • Chapter 60. Cardinals

XVI. Set Theory II: Model Theory and Forcing

  • Chapter 61. Inner Model Theory
  • Chapter 62. Forcing
  • Chapter 63. Breaking the Continuum Hypothesis

http://web.evanchen.cc/FAQs/contest.html

Math Contest FAQ's

Return to FAQ Index.

These are FAQ's about math contests and particularly how to go about training for them.

 

How should I prepare for math contests?

One-sentence answer: do lots of problems just above your current ability, and spend some of time on reflection. Full answer: this blog post.

It's worth pointing out that no one has a silver bullet: there's no known study method X (let alone a specific handout/book) such that students following X consistently do well on USAMO. (Reason: the existence of X is not a Nash equilibrium.) There are plenty of strategies which obviously don't work like "do nothing" and "read solutions without trying any problems", but beyond that anything reasonable is probably fine.

If you had to force me to say what I thought was the biggest predictor of success, I would say it iswhether you think about math in the shower. That means you both enjoy your work and are working on things in the right difficulty range.

 

Which books/handouts/materials should I use?

As I said it probably doesn't matter too much which ones you choose, as long as you exercise some basic common sense.

With that disclaimer, here are some possible suggestions for math olympiads.

Younger students (preparing for AMC/AIME) would likely benefit from books or classes from Art of Problem Solving, like Volume 2.

 

How do I get better at Euclidean geometry?

For general advice, see my advice, Geoff Smith's advice, etc.

For specific materials, see my own handouts or Yufei Zhao's, among others. And of course, my geometry textbook.

 

Is it possible for me to go from X level to Y level in Z time?

I probably don't know much more than you do.

Predicting improvement in math contests is a lot like trying to predict the stock market. We have some common sense, but no one really knows much more than that.

 

How do I learn to write proofs?

I don't think there's actually a leap between computation and proof-writing, and I actually suspect that thinking proof-writing is hard is most of what makes it hard. Reasons it might appear hard:

  • US students get basically zero exposure to proofs (even in the contest world), and are led to think that it's something mythical that's above them, when in fact it's merely just being forced to explain your solution on paper.
  • Having zero experience also means that you might not present your ideas clearly or violate some unspoken rules on style.
  • Some classes of problems (like inequalities) don't appear at all until the olympiad level, so students have to learn how to write a proof (fairly easy) while simultaneously learning a new class of problems (hard).

To get you started, try reading this article, or perhaps this article. After this, my advice is

  1. Read the proofs to problems you think you've solved (on AoPS, or official solutions). Note that these don't have to be from proof contests! The official solutions to any decent contest would all pass as proofs.
  2. Try writing up proofs to problems you think you've solved, and
  3. Get feedback on these solutions (from a mentor, on the forums, etc.).

The USA Mathematical Talent Search is also another good option. It is a free mathematics competition open to all United States middle and high school students, which gives you about one month to produce full solutions to a set of five problems.

 

Am I ready to do X level problems, read book Y, etc.?

The correct thing to do is just try it out (e.g. try some problems from a past X paper, read a chapter from Y, etc.) and see how it feels. You all know what it feels like when something is too easy (think middle school math class): you feel like you're doing the problems for the sake of doing the problems rather than actually learning. You all know what it feels like when something is too hard: the dreaded "I have no clue what's going on".

Anything not too close to either extreme is probably fine. If in doubt, I recommend picking whatever is most enjoyable.

For what it's worth, I think most students are too conservative in doing harder problems. Just because you haven't qualified for USAMO yet doesn't mean you can't try some USAMO 1/4's!

 

Should I read X book versus Y book, spend M hours versus N hours per week, etc.?

My gut feeling is that the effect size of this is sufficiently small that (i) no one knows a definitive answer, (ii) the answer is likely to depend on the person, and (iii) it's rounding error compared to the actual final result. Therefore I think the correct thing to do is try both, see which one you like better, and just go with that, without worrying about whether it is "right".

 

How do I make fewer careless errors?

There's a nice article on AoPS that addresses most of what I have to say. Here are just a few additional remarks.

We're not kidding when we say to be neat: it really is hard to think if your scratch paper is a bunch of clutter. To get an idea of what my scratch work looks like, here is my scratch paper from the 2013 AIME. Some things worth noting from it are:

  • Every problem is labelled on its own page (or multiple pages).
  • Diagrams are very large, often taking up half the page.
  • Mistakes are simply "struck out" rather than scribbled out or halfheartedly erased.
  • The final answer on each page is boxed for easy reference later.

Also, don't misread questions, don't rush, etc.

Anyways, I admit there's not always a whole lot you can do about it. 2013 was the first year where I was able to look at a problem and basically know how to do it within one or two minutes; this left me a lot of time for computation, and consequently I made very few errors as compared to 2011 or 2012. In other words, as you get better at problem-solving you'll naturally become less likely to make careless errors as well. (At least that's how it turned out for me.)

See also: Against Perfect Scores.

 

Can you solve X problem for me?

Probably not. If you send me a problem, usually I will at least read it. If I have seen it before or can quickly see how to do it, I will generally be nice enough to write back and outline or link the solution. But otherwise I will likely be too embarrassed to admit I don't have time to work on every problem that students send me, and simply archive your message.

 

http://www.iarcs.org.in/inoi/online-study-material/topics/

Indian Computing Olympiad

Basic topics

 

Advanced topics

http://www.iarcs.org.in/inoi/online-study-material/problems/

Searching

 

 

Sorting

 

 

Basic Graph Algorithms

 

 

Dynamic Programming

 

 

Greedy Algorithms

 

What this page is about

This webpage contains online self-study material for students preparing for the Indian Computing Olympiad. The Computing Olympiad is a test of knowledge and skill in algorithms and programming. Teaching programming is beyond the scope of this training material. The focus is on presenting some basic ideas about algorithms and data structures, primarily through representative problems.

Practice problems

Some of the problems discussed in these notes, and many others that require similar techniques, can be found at the the The IARCS Problems Archive. This is an online resource where you can submit your code and have it automatically evaluated on representative test cases. Other, similar, sites are listed on the left under Online judges.

Reading material

The best place to learn the background material is a good textbook on algorithms.

A good book that combines detailed presentations with many interesting examples is the following.

Algorithm Design, by Jon Kleinberg and Evá Tardos

 

This next book is probably the most comprehensive textbook available on algorithms, but it can also be a bit heavy to read.

Introduction to Algorithms, by Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest and Clifford Stein

 

An easy-to-read and accessible book is the following one. It used to be freely available online, but longer is, unfortunately.

Algorithms, by S. Dasgupta, C.H. Papadimitriou, and U.V. Vazirani

 

http://web.evanchen.cc/problems.html

USA Team Selection Tests

On this page I maintain archives of the USA team selection tests, the ELMO.

Note that this is not an official repository of problems and solutions, nor do I own the copyright for any of the problems. In fact, many of the files are replicas of the originals produced by me. However, all these problems and solutions can be found on Art of Problem Solving anyways, I have assembled a subset of them here for everyone's convenience.

USA TST Selection Test (TSTST)

For an explanation of the name, see the FAQ on the USA IMO team selection.

USA Team Selection Test (TST)

ELMO

Okay, this isn't actually a USA team selection test, but I couldn't resist.

Taiwan Team Selection Test

This is not a USA team selection test either, but I also couldn't resist. These problems are in Chinese;English versions here.

 

 

 

 

 

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