2002 Boston USAMO Event Pictures
boopsey03 photos
Tips for Olympiad participants
kedlaya@math.mit.edu
The term "olympiad" is used generically to refer to a math contest in which students are asked not to compute numerical answers, but to give proofs of specified statements. (Example: "Prove that 2003 is not the sum of two squares of integers.") The most famous example is the International Mathematical Olympiad; most countries that participate at the IMO have national olympiads as part of their team selection process. Some areas have additional olympiads at the regional or local level.
The jump from short answers to olympiads is a tough one. Here are some tips for students making this transition.
- Practice, practice, practice. The only way to learn math is by doing.
- Proofs are essays. The better written a proof is, the more likely it is to be understood. Even such mundane things as grammar, spelling and handwriting are worth a bit of attention.
- Define your terms. If you're going to use a word in a way that might not be commonly understood, define it precisely. Then stick to your definition!
- Read the masters. No one ever learned how to do good mathematics in a vacuum. When you do practice problems, read the solutions even of the problems you solved.
- There's more than one road. Different solutions can be equally valid; even when solutions agree in substance, differences in perspective can be significant and valuable.
- It's not over when it's over. Don't hesitate to continue thinking about the problems on a contest after the time ends, or to discuss the problems with others.
- Learn from your peers. They're smarter than you might have expected.
- Learn from the past. Try to relate new problems to old ones; you may learn something from the similarities, or from the differences.
- Patience. No one said this was easy!
If you want to do well on Olympiad-style math contests (those requiring essay-type answers with full explanations and rigorous proofs), you had better do your homework. You'll learn some interesting mathematics in the process! Listed by Kiran S. Kedlaya (kedlaya@math.mit.edu) for American Mathematics Competitions.
Sources of problems
Olympiad books/booklets
The Mathematical Association of America publishes two volumes of problems from the International Mathematical Olympiad (IMO) and one from the USA Mathematical Olympiad (USAMO). These three books are part of the MAA New Mathematical Library (NML) series.
However, the IMO books only cover problems up to 1985, and the USAMO book up to 1986. (I'm told subsequent books in each series are in the works, but don't expect them any time soon.) There are two ways to remedy this problem. First, the American Mathematics Competitions (AMC) makes the problems of these missing Olympiads available on their Web site. Second and preferable for Olympiad preparation, the AMC also publishes one pamphlet for each year, containing the USAMO and IMO problems from that year with detailed solutions. See the AMC web site for order information.
Three collections of national olympiads are also available from AMC, covering 1995-1996, 1996-1997, and 1997-1998. Each book includes problems and solutions from the prior year, and problems from the latter year. When ordering, ask for "Mathematical Contests".
Other problem books
There are lots of other sources of Olympiad-style problems. The USSR Olympiad Problem Book by Shlarsky, Chentzov, Yaglom is but one example.
Perhaps more useful than books of problems are books that also include some comments on problem-solving. A good choice is the recent A Mathematical Mosaic, by former Canadian IMOer Ravi Vakil.
Before his death, Samuel Greitzer (founder of the USA IMO team) published a journal for students called the Arbelos, containing a mixture of problems and commentary. The AMC sells copies of this journal in 6 bound pamphlet.
Journals with problem sections
The journal Crux Mathematicorum is devoted entirely to Olympiad-style problem solving, and is read by many IMO hopefuls around the world.
Many other mathematics journals have a regular problem section, featuring problems and solutions submitted by readers. Two American journals whose problems are suitable for Olympiad participants are the MAA's Mathematics Magazine and the American Mathematical Monthly.
Don't forget, mathematics journals for high school students exist in many countries. For example, Bulgaria has Matematika, Hungary has Kömal, and Russia has Kvant (also published in the U.S. as Quantum).
Background reading on particular topics
While much of the material on this list is intended to be directly relevant to competitions, the suggestions should also be useful to those wishing to study these topics for their own sake.
Algebra
Polynomials, by Barbeau, is a good starting point.
If abstract algebra (groups, rings, fields, Galois theory) is what you're looking for, Contemporary Abstract Algebra by Gallian is a low-impact introduction to the subject. More sophisticated texts include Algebra, by Artin and Topics in Algebra, by Herstein.
Combinatorics
This subject is blessed by an abundance of well-written texts. A good source for enumerative combinatorics is Richard Stanley's book of the same name (2 volumes). For generating functions, look no further than generatingfunctionology, by Wilf. Concrete Mathematics, by Graham, Knuth and Patashnik, is not easy reading for the beginner, but includes a slew of problems.
Game Theory
The bible of this subject is Winning Ways for your Mathematical Plays, by Berlekamp, Conway and Guy (two volumes).
Geometry
The standard supplement for the American student underprepared in Euclidean geometry is Geometry Revisited, by Coxeter and Greitzer (part of the MAA's NML series). Additional reading could include Geometric Transformations, by Yaglom (NML, 3 volumes) and/or A Course in Geometry, by Eves (2 volumes).
For those who want to pursue geometry in some of its more modern incarnations, here are a couple of additional suggestions. A gentle introduction to hyperbolic geometry is Journey into Geometries, by Sved (MAA). For projective geometry, try Coxeter's book of the same name.
Graph Theory
Try Pearls in Graph Theory by Hartsfield and Ringel.
Inequalities
The bible of this subject is Inequalities, by Hardy, Littlewood and Pólya, but it makes for pretty tough reading. Better would be to start with my MOP 1998 notes (see the resources for students page).
Number Theory
Try Vanden Eynden for beginners, Sierpinski for more advanced readers. The ambitious student might try Niven and Zuckerman.
For those who want to pursue the subject further, try An Introduction to the Theory of Numbers, by Hardy and Wright; A Classical Introduction to Modern Number Theory, by Ireland and Rosen, The Theory of Algebraic Numbers, by Diamond and Pollack; or Number Fields, by Marcus.
Other stuff
TeX
TeX (pronounced like the "tech" in "technology") is the standard system for typesetting mathematics. One constructs mathematical symbols and equations by describing what they say, rather than what they look like, in human-readable code that is easily transmitted over the Internet. TeX is much more powerful than most commercially available systems, and is freely available to boot!
There are lots of books about TeX (and LaTeX, an enhanced version of plain TeX) out there, and some are quite good. You might try The LaTeX Companion, by Goossens et al; or LateX: A Document Preparation System, by Lamport.
Some On-Line sites related to math
Art of Problem Solving http://www.artofproblemsolving.com
A site developed to help students learn how to solve the puzzling problems --
The creators of AoP were this student once. They were the kids who wanted to win the trophies. They worked hard and became the kids who won the trophies. The trophies are in attics now. The problem-solving skills, the love of mathematics, and the friendships forged with peers with similar interests remain. They've applied the skills we've developed through mathematics to a variety of fields in college, then in the professional world.
"Now we've returned to our starting point - the student in a room, chewing on a pencil, staring at a question, giving up, reading the answer, and thinking. . . How would I have thought of that?
"This time you are the student. We are building this site for you, to provide a resource you can turn to.
"You're stuck on a problem, so you write friends on our Forum. You hang out in our Math Jams. You take an online class. You don't give up. You learn how to think of the solution. You solve the problem. Then you think..........Next problem.
Awesome Math http://www.awesomemath.org/
AwesomeMath consists of three major intiatives.
The AwesomeMath Summer Program is a three-week camp designed to hone high school students' mathematical problem-solving skills up to the Olympiad level.
The AwesomeMath Year-round program is an effort to continue students' enrichment during the school year through a series of correspondence lectures and problem sets.
Mathematical Reflections is an online journal that presents students with quality mathematical writing and gives them the opportunity to formally publish their own exceptional work.
Calculus the Musical http://lwww.calculusthemusical.com
Matheatre is an educational performance duo.-- touring our original production of "Calculus: The
Musical!," a comic "review" of the concepts and history of Calculus. Created by a licensed math teacher and a professional theatre artist, Matheatre strives to put the "edge" back in "education!" For services contact
us at calculusthemusical@gmail.com. All materials on our website are free for use by any student, educator
or fan of mathematics!
Circus of Patterns http://www.circusofpatterns.com/
This site is a product of a mathematicians research in mathematics number patterns for 25 years. He has developed a series of mathematical charts for teachers and those students who do not like math, so they become involve in mathematics and having "FUN". The series of math charts are in, whole number, fraction, and decimals. Students are draw to the board searching for numerical and geometric patterns, as they are having "FUN" to doing the math.
Cut the Knot http://www.cut-the-knot.org
The site, among others, is the winner of the MERLOT Classics 2004 award and a 2003 selection of the Scientific American. It's a big site with more than 500 Java illustrations. Topics covered are drawn from Arithmetic, Algebra, Geometry, Probability, Calculus, Social Sciences, Logic and more.
Global Institute of Mathematics http://www.globalmath.org/
GIOM offers highly interactive courses from Algebra through Calculus in real time. Students can interact with the instructor as lessons are taking place. Lessons are also archived for future viewing. Students have unlimited email access to the instructor outside of lesson time.
IMO - Official Site http://www.imo-official.org/
There is a lot of information from the previous IMOs on this site, but a lot of the data is still missing. Please take some time and send the missing results for the competitors from your country to the webmaster
( webmaster@imo-official.org
IMO Compendium http://www.imo.org.yu
This website is dedicated to mathematical olympiads, and we hope it will be of use to all those who prepare for math competitions or simply love problem mathematics.
International Mathematics Project Competition (IMPC) http://www.katev.org/impc
We honorably inform that the seventh International Mathematics Project Competition (IMPC-2005) will be held between 17-21 May 2005 in Almaty, the old capital city of Kazakhstan. The competition aims at inspiring and motivating mathematically talented high school students by exposing them to the beauty and variety of mathematics with technological application.
Kiran Kedlaya's Math Related Web Sites Listing http://unl.edu/amc/a-activities/a4-for-students/K-links.html
This directory is intended to catalog resources on the Web of possible use to mathematically motivated students, their parents and teachers.
Math Forum http://mathforum.com/mam/00/612/index.html
This page was designed for Mathematics Awareness Month. It is ranked as one of the best. It contains links to mathematicians who have made significant contributions to mathematics and its applications.
MathCounts http://www.mathcounts.org
A national math competition for Jr. High Students (6-8th graders) -- it has been around for more than 20 years. Is held regionally, state wide, and then nation wide.
Math Is Power http://www.mathispower.com/
Contains Problems of the Week relating elementary and middle school mathematics in geometry, algebra, discrete mathematics, trigonometry and calculus.
Math Meet http://math.uww.edu/mathmeet
A free, on-line, team mathematics competition for middle and high school teams. Held Mid April, sponsored by CLARC.
MathPath http://www.mathpath.org/
Online learning resource and summer camp for grades 6, 7, and 8.
Nick's Mathematical Puzzles http://www.qbyte.org/puzzles/
A collection of more than 100 puzzles ranging over geometry, probability, number theory, algebra, calculus, and logic. Hints are provided, along with answers, fully worked solutions, and links to related mathematical topics. Many of the puzzles are elementary in their statement, yet challenging. New puzzles are added on a regular basis.
Online Companion to the Special Interfaces Issue http://wwwlinforms.org/ebiz/interfaces
A forthcoming special issue of Interfaces on OR/MS applied to e-business is now freely available. Its seven papers collectively demonstrate that decision technology is becoming a powerful adjunct to information technology as the digital economy matures.
QuestBridge http://www.questbridge.org
Pairs outstanding low-income stuents with full four-year scholarships to 12 QuestBridge partner colleges. Qualified students can apply for admission to any or all of our partner schools through the College Match using our free online application.
Rasko Jovanovic`s World of Mathematics http://milan.milanovic.rog/math/english/contests.html
Explores a connections between Pascal Triangle and the Fibonacci numbers.
Swarthmore http://forum.swarthmore.edu
A must page for students and teachers of mathematics K-12. This site contains many useful links and features a Problem of the Week.
Terry Wesner's page http://www.totallyfreemath.com
An author for William C. Brown/McGraw Hill Publishers for 25 years. "I feel it is time to start giving back to the educational community that has supported me for all of those years. As time and resources allow, I will be providing all of my books and the teacher's resource materials for free download.
The site
THEA practice tests http://www.testprepreview.com/thea_practice.htm
These are online THEA (Texas Higher Education Assessment) practice tests at no charge. The site in general also has a wide variety of other practice tests.
Valentin Vornicu http://www.mathlinks.ro
A former Romanian IMO Team Member, currently an undergraduate student in Mathematics at University of Bucharest. He thought that the site he had forged would fit in quite nicely.
The site's address would be www.mathlinks.ro and it has a math forum, with a lot of users, most of them being former, current or future IMO contestants, download problems section and also a weekly contest - similar with USAMO in difficulty. It is not as big as the mathforum (yet :-) ).
Wolfram http://mathworld.wolfram.com/
A free service for the mathematical community provided by Wolfram Research, makers of Mathematica, with additional support from the National Science Foundation Includes all sorts of problem solving pages.
World of Mathematical Equations http://eqworld.ipmnet.ru
The EqWorld website presents information on solutions to various classes of algebraic, ordinary differential, partial differential (mathematical physics), integral, and other mathematical equations. It also outlines some methods for solving equations, includes interesting articles, lists, useful handbooks, and "monographs," etc. More advanced than most high school math levels, but included here for reference.
Kiran Kedlaya -- Math Related Web Sites
This directory is intended to catalog resources on the Web of possible use to mathematically motivated students, their parents and teachers.
The directory was started by Kiran Kedlaya (http://www-math.mit.edu/~kedlaya/)
(kedlaya[at]math[dot]mit[dot]edu). Please contact me directly with comments about broken links or suggested new links
Math circles
A "math circle" is a group of students and adults who get together periodically to explore mathematics in an informal, extracurricular setting. The phenomenon seems to have begun in the Soviet Union (as described in the book Mathematical Circles, by Fomin, Genkin, and Itenberg), but has since been brought to America by a wave of expatriate Russian mathematicians.
A similar function is often served by the practice meetings of teams for the ARML competition. However, math circles usually involve deeper exploration and less emphasis on competitions.
Math circles, as well as other programs that do not strictly follow the "circle" protocol but operate in a similar spirit, are actually pretty widespread; if you live in an at all populated area of the US (especially near a major university), there probably is one near you. (More links to such programs would be most appreciated!)
The Art of Problem Solving folks apparently run a group at the University of California, San Diego.
Bob Kaplan's math circle (Boston area)
Bay Area Mathematical Circles consists of four circles in the San Francisco area. I have been involved with one of these, the Berkeley Math Circle.
The Mu Alpha Theta organization is a sort of national analogue of what I'm describing here. (That and I couldn't think of where else to put it.)
University of Chicago Young Scholars Program
UC Irvine Math Circle
University of Texas Saturday Morning Math Group
University of Utah Math Circle
For middle school students
This section is out of my expertise, so I could use suggestions for useful links.
Art of Problem Solvingis a new discussion forum; it may also be relevant for high schoolers.
Mathcounts
Johns Hopkins' Study of Exceptional Talent (the original SAT-at-age-13 program, affiliated with the CTY summer study program) has descendants in various parts of the country, e.g., Duke's Talent Identification Program.
Math is Fun is mainly a K-12 website, but it can sometimes help to have the basic concepts explained, plus we have plenty of math puzzles. The site has been around for 6 years, and is popular with schools.
Visual Math Learning is a free online interactive tutorial for pre-algebra students that is rich in games, puzzles, and animated manipulatives that emphasize learning concepts by visualization
Kodawari House mathematics, including arithmetic, created for children.
Regional contests
Many of these links were harvested from this site.
Bay Area Math Meet (San Francisco area)
Bay Area Math Olympiad(San Francisco area)
Colorado Math Olympiad
Florida Atlantic University-Stuyvesant Alumni Mathematics Competition (Palm Beach/Broward counties and vicinity)
Furman University Wylie Mathematics Tournament (Southeast US)
Harvard-MIT Math Tournament (New England)
Lehigh University High School Math Contest (Pennsylvania)
Maritime Mathematics Competition (Canadian maritime provinces)
Massachusetts Association of Math Leagues
Nassau Country Interscholastic Math League (Long Island)
The Math League (New England)
North Carolina State Math Contest
Oklahoma State University High School Math Contest
Polya Mathematics Competition (San Francisco area)
Rice Math Tournament (Houston)
University of Maryland Math Competition
University of South Carolina High School Math Contest
University of Wisconsin Talent Search
National contests
Beware that "national" sometimes means US, sometimes Canada, sometimes both. For international contests, see the AMC Problem Directory and/or the IOI Secretariat.
AMATYC Student Math League
American Mathematics Competitions
American Regions Math League; includes links to team web sites
Canadian Mathematics Competition
Canadian Mathematical Olympiad
Canadian Open Mathematics Challenge
IML Math League
International Mathematical Talent Search
Mandelbrot
Mathematical Contest in Modeling
Mu Alpha Theta
USA Math Talent Search
Science fairs
These are competitions in which students submit research projects that they have been working on for some time beforehand. Usually these projects are done either in collaboration with, or at least at the suggestion of, a mentor; the RSI program specializes in connecting students with mentors and projects, but you may be able to find one on your own simply by getting in touch with, say, someone at your local university.
Davidson Fellowships (arts and sciences)
Intel (formerly Westinghouse) Science Talent Search (seniors only)
International Science and Engineering Fair
Siemens Westinghouse Science and Technology Competition (seniors only)
Summer (and other times) programs
A number of these programs are supported by the American Mathematical Society Epsilon Fund, which is actively seeking contributions to build an endowment. The AMS also maintains a directory of summer math programs more comprehensive than this one.
Canada/USA Mathcamp (various locations)
Clay Mathematics Institute Research Academy (Boston; not summer)
Hampshire College Summer Studies in Mathematics
Math Olympiad Program (University of Nebraska)
PROMYS (Boston University)
Ross Young Scholars Program (Ohio State University)
Research Science Institute (MIT)
Rutgers Young Scholars Program
Southwest Texas Honors Summer Math Camp
Stanford Mathematics Camp
University of Michigan Math Scholars
Journals, newsletters, and other correspondence
Crux Mathematicorum with Mathematical Mayhem
KöMaL
Mathematical Olympiads Correspondence Program
Suggestions for further inquiry
Crux Mathematicorum with Mathematical Mayhem downloadable
pictures of the 2006 MOSP
The Akamai Foundation
The Akamai Foundation was established by Akamai Technologies, Inc. and was initially funded by Akamai executives and its employees. Akamai Technologies helps companies by optimizing Web site performance, delivering broadcast-quality streaming media, and providing interactive application services. The Foundation chose to focus on mathematics because Akamai was conceived and founded on mathematical innovation.
The Akamai Foundation is dedicated to excellence in mathematics in the hopes that we can encourage America's next generation of technology innovators. Their mission is to reach out to students in grades K-12 with the message that mathematics is important, demonstrating to them that it can be magical and fun and in the process, helping them to realize that math can lead to some very exciting career opportunities.
In February 2001, just as the AMC 10 and AMC 12 were being given and the AIME 2001 was going to press, the Akamai Foundation of Boston made a large charitable gift to the Mathematical Association of America (MAA) on behalf of the American Mathematics Competitions. This large charitable gift allowed MAA American Mathematics Competitions to fund scholarships in 2001 and 2002, and a national USAMO in 2002.
In 2001and 2002 the Akamai Foundation provided $1,000 scholarships to the top male and female American Invitational Mathematics Examination (AIME) scores in each state.
During the 2001-2002 academic year:
The AIME scholarships for the top scorers from each state continued
Participants in the United States of America Mathematical Olympiad (USAMO) were flown in May 2002 to MIT (Massachusetts Institute of Technology), Cambridge, MA, for the administration of the Olympiad. Through its charitable gift, The Akamai Foundation served as host to the students, proctors, and exam graders for this three day event. Students were also treated to area tours, special guest speakers, and related social and mathematical activities.
In June 2002, there was an enhanced Mathematical Olympiad Summer Program (MOSP), expanding the number of students to 178.
In 2004 the Akamai Foundation provided additional funds so the summer MOSP program could include an additional 24 students (9th graders).
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