IMO

IMO

 

TRADUZINDO, CHECANDO AS INFORMAÇÕES E CONSERTANDO OS LINKS

fonte: https://olympiads.win.tue.nl/imo/index.html

 

More about IMO:
HARD PROBLEMS: The Road to the World's Toughest Math Contest

The International Mathematics Olympiad (IMO, also known as the International Mathematical Olympiad) is an annual mathematics competition for high school students [IMO Article in Wikipedia]. It is one - in fact, the oldest - of the International Science Olympiads. The first IMO was held in Romania in 1959. The problems come from various areas of mathematics, such as are included in math curricula at secondary schools. Finding the solutions of these problems, however, requires exceptional mathematical ability and excellent mathematical knowledge on the part of the contestants.

Topics covered (see Unwritten Syllabus and Tutorials at Arkadii Slinko's Math Olympiad Website, currently being transferred):

  • Number Theory, including
    • Fundamental Theorems on Arithmetic
    • Linear and quadratic Diophantine equations, including Pell's equation
    • Arithmetic of residues modulo n, Fermat's and Euler's theorems
  • Algebra, including
    • Fundamental Theorems on Algebra, e.g. inequalities, factorization of a polynomial into a product of irreducible polynomials
    • Symmetric polynomials of several variables, Vieta's theorem
  • Combinatorics, including
    • Graph theory
  • Geometry, including
    • Properties of the orthocentre, Euler's line, nine-point-circle, Simson line, Ptolemy's inequality, Ceva and Menelaus etc.

Excluded topics:

  • Calculus (!)
  • Complex numbers (though present in the past)
  • Inversion in geometry
  • Solid geometry (though present in the past; may return)

The usual size of an official delegation to an IMO is (a maximum of) six student competitors and (a maximum of) two leaders. There is no official ``team''. The student competitors write two papers, on consecutive days, each paper consisting of three questions. Each question is worth seven marks. (The preceding information is taken from an Overview of the IMO provided by the IMO'95 host country, Canada; also see below.) A total score of 42 points is possible. Awards are determined as follows:

  • GOLD MEDAL: the top 1/12 of scores receive gold medals
  • SILVER MEDAL: the next 2/12 of scores receive silver medals
  • BRONZE MEDAL: the next 3/12 of scores receive bronze medals
  • HONORABLE MENTION: any competitor who receives a perfect score of 7 on any one question, but who does not receive a medal, is awarded an honorable mention

My report on IMO 2002 provides further details of how the IMO is run.

 

Estudo

 

 
Andy Liu, Bruce Shawyer. Problems from Murray Klamkin
Martin Erickson. Aha! Solutions
Zvezdelina Stankova, Tom Rike. A Decade of the Berkeley Math Circle
Sam Vandervelde. Circle in a Box
Alexander Zawaira, Gavin Hitchcock. A Primer for Mathematics Competitions
Valentin Boju, Louis Funar. The Math Problems Notebook
Béla Bollobás. The Art of Mathematics: Coffee Time in Memphis
Titu Andreescu, et.al. 103 Trigonometry Problems
Titu Andreescu, et.al. 104 Number Theory Problems
Titu Andreescu et al. Mathematical Olympiad Challenges
Titu Andreescu et al. Mathematical Olympiad Treasures
Terence Tao. Solving Mathematical Problems: A Personal Perspective
Jiri Herman et al. Equations and Inequalities
Jiri Herman et al. Counting and Configurations
B. J. Venkatachala. Functional Equations: A Problem Solving Approach
Christopher Small. Functional Equations and How to Solve Them
Publications by the UK Mathematics Trust
Dmitry Fomin, et al. Mathematical Circles: Russian Experience
Robert and Ellen Kaplan. Out of the Labyrinth: Setting Mathematics Free
Alfred Posamentier, et.al. Problem-Solving Strategies For Efficient and Elegant Solutions
Douglas Faires. First Steps for Math Olympians
Michael Steele. The Cauchy-Schwarz Master Class: An Introduction to the Art of Inequalities
Loren Larson. Problem Solving Through Problems
Bonnie Averbach, Orin Chein. Problem Solving Through Recreational Mathematics
Liong-shin Hahn. New Mexico Mathematics Contest Problem Book
William Briggs. Ants, Bikes, and Clocks: Problem Solving for Undergraduates
Steven Krantz. Techniques of Problem Solving
Wayne Wickelgren. How to Solve Mathematical Problems
Dusan Djukic. The IMO Compendium
Jörg Bewersdorff. Luck, Logic, and White Lies
Steve Olson. Count Down
David Acheson. 1089 and All That: A Journey into Mathematics.
Paul Zeitz. The Art and Craft of Problem Solving

Problem Solving

The classic book about solving mathematical problems is:

  • How to Solve It: A New Aspect of Mathematical Method.
    G. Polya
    Second Edition, Princeton University Press, 1957. [See this book at Amazon.com]

A kind of sequel to Polya's How to Solve It, presenting modern heuristics, especially for bigger problems (math problems, not `just' programming problems) requiring computers:

  • How to Solve It: Modern Heuristics (2nd Ed.).
    Z. Michalewicz and D. B. Fogel.
    Springer, 2004 (First Edition 2000). [See this book at Amazon.com]

Book Cover Another book that will help you become a good math problem solver, by distinguishing `mere' exercises from (challenging, unpredictable) real problems (the author participated in IMO 1974):

Excellent IMO training material:

A good initial preparation for IMO-style problem solving:

  • A Primer for Mathematics Competitions. =====NEW IN LIST=====
    Alexander Zawaira, Gavin Hitchcock. Oxford University Press, 2009.
    [See this book at Amazon.com]

     

More training material:

From a 1988 IMO gold-medal winner, summarizing Polya's method and applying it to 26 diverse contest problems:

More aimed at beginners, and especially their teachers:

  • Problem-Solving Strategies For Efficient and Elegant Solutions: A Resource for the Mathematics Teacher.
    Alfred S. Posamentier, Stephen Krulik.
    Corwin Press, 2003. [See this book at Amazon.com]

Based on the various (levels of) USA mathematical competitions:

  • First Steps for Math Olympians: Using the American Mathematics Competitions.
    J. Douglas Faires.
    MAA, 2006. [See this book at Amazon.com]

Approaching problem solving via puzzles and games:

  • Problem Solving Through Recreational Mathematics.
    Bonnie Averbach, Orin Chein.
    Dover Publications, 1999. [See this book at Amazon.com]

Another general overview with many sample problems:

Another classic on mathematical problem solving, adding a psychological perspective:

  • How to Solve Mathematical Problems.
    Wayne A. Wickelgren.
    Dover Publications, 1995. [See this book at Amazon.com]
    This is a corrected republication of How to Solve Problems: Elements of a Theory of Problems and Problem Solving, Freeman & Co., 1974.

Collections of Mathematical (Competition) Problems

There are many collections of mathematical (contest) problems (incl. IMO):

  • The IMO Compendium: A Collection of Problems Suggested for The International Mathematical Olympiads: 1959-2004.
    Dusan Djukic, Vladimir Z. Jankovic, Ivan Matic, Nikola Petrovic.
    Springer, 2005. [See this book at Amazon.com]
    Companion website

     

  • The Art of Mathematics: Coffee Time in Memphis.
    Béla Bollobás.
    Cambridge University Press, 2006. [See this book at Amazon.com]
    Not all topics are relevant for the IMO, but it sure is a nice broad intriguing collection of some 157 problems, with hints and solutions.
  • 102 Combinatorial Problems : From the Training of the USA IMO Team.
    Titu Andreescu, Zuming Feng.
    Birkhäuser, 2003. [See this book at Amazon.com]

     

  • 103 Trigonometry Problems : From the Training of the USA IMO Team.
    Titu Andreescu, Zuming Feng.
    Birkhäuser, 2004. [See this book at Amazon.com]

     

  • 104 Number Theory Problems : From the Training of the USA IMO Team.
    Titu Andreescu, Dorin Andrica, Zuming Feng.
    Birkhäuser, 2006. [See this book at Amazon.com]

     

  • Books on mathematical problem solving at the Australian Mathematics Trust (AMT); here is a small selection:
    • Seeking Solutions : Discussion and Solutions of the Problems from the International Mathematical Olympiads 1988-1990.
      J. C. Burns.
      AMT, 2000.
      [See this book at Amazon.com]

       

    • 101 Problems in Algebra : From the Training of the USA IMO Team.
      Titu Andreescu, Zuming Feng.
      AMT, 2001.
      [See this book at Amazon.com]

       

    [End of selection from AMT]
  • Publications on mathematical problem solving at the UK Mathematics Trust (UKMT); here is a small selection (I have no direct links to Amazon.com):
    • Introductions to Number Theory and Inequalities.
      J. C. Burns.
      UKMT, 200?.
      ISBN 0-9536823-8-2

       

    • Plane Euclidean Geometry.
      A. D. Gardiner and C. J. Bradley.
      UKMT, 2005.
      ISBN 0-9536823-6-6

       

    [End of selection from UKMT]
  • Mathematical Puzzles: A Connoisseur's Collection.
    Peter Winkler.
    A. K. Peters, 2004. [See this book at Amazon.com]

     

  • The Math Problems Notebook. =====NEW IN LIST=====
    Valentin Boju, Louis Funar.
    [See this book at Amazon.com]
  • Ants, Bikes, and Clocks: Problem Solving for Undergraduates.
    William Briggs.
    SIAM, 2004.
    [See this book at Amazon.com]
  • Books on problem solving published by the Mathematical Association of America (MAA); here is a small selection:
    • International Mathematical Olympiads, 1955-1977.
      Samuel L. Greitzer (ed.).
      MAA, 1979. [See this book at Amazon.com]
    • International Mathematical Olympiads, 1978-1985, and Forty Supplementary Problems.
      Murray S. Klamkin (ed.).
      MAA, 1986. [See this book at Amazon.com]
      This book includes tables that list participating countries and their performance at IMOs from the beginning in 1959 to 1985.
    • International Mathematical Olympiads, 1986-1999.
      Marcin E. Kuczma (ed.).
      MAA, 2003.
      [See this book at Amazon.com]
    • USA Mathematical Olympiads 1972-1986 Problems and Solutions.
      Murray S. Klamkin (ed.).
      MAA, 1989. [See this book at Amazon.com]
    • Five Hundred Mathematical Challenges.
      Edward J. Barbeau, Murray S. Klamkin, William O. J. Moser.
      MAA, 1995. [See this book at Amazon.com]
      Includes solutions. From the preface: ``This collection of problems is directed to students in high school, college and university. Some of the problems are easy, needing no more than common sense and clear reasoning to solve. Others may require some of the results and techniques which we have included in the Tool Chest [15 pages with results from Combinatorics, Arithmetic, Algebra, Inequalities, Geometry and Trigonomotry, and Analysis]. None of the problems require calculus... They could be described as challenging, interesting, thought-provoking, fascinating.''
    • From Erdos to Kiev : Problems of Olympiad Caliber.
      Ross Honsberger.
      MAA, 1996. [See this book at Amazon.com]
    • In Pólya's Footsteps: Miscellaneous Problems and Essays.
      Ross Honsberger.
      MAA, 1997. [See this book at Amazon.com]
    • Mathematical Miniatures.
      Svetsoslav Savchev and Titu Andreescu.
      MAA, 2003. [See this book at Amazon.com]
    • Problems from Murray Klamkin. =====NEW IN LIST=====
      Andy Liu, Bruce Shawyer (Eds.).
      MAA, 2009. [See this book at Amazon.com]
    • Aha! Solutions. =====NEW IN LIST=====
      Martin Erickson.
      MAA, 2009. [See this book at Amazon.com]
    [End of selection from MAA.]
  • Mathematical Olympiad Challenges, Second Edition.
    by Titu Andreescu and Razvan Gelca.
    Birkhäuser Boston, 2000.
    [See this book at Amazon.com]
  • Mathematical Olympiad Treasures.
    by Titu Andreescu and Bogden Eneescu.
    Birkhäuser Boston, 2003.
    [See this book at Amazon.com]
  • Winning Solutions.
    Edward Lozansky and Cecil Rousseau.
    Springer-Verlag, 1996. [See this book at Amazon.com]
    From the preface: ``There is a significant gap between what most high school mathematics programs teach and what is expected of an IMO participant. This book is part of an effort to bridge that gap.''
    Chapters on Numbers, Algebra, and Combinatorics. Explains some theory, provides examples, exercises and solutions.
  • Colorado Mathematical Olympiad: The First Ten Years and Further Explorations.
    Alexander Soifer.
    Center of Excellence in Publ., 1994. [See this book at Amazon.com]
  • New Mexico Mathematics Contest Problem Book.
    Liong-shin Hahn.
    University of New Mexico Press, 2005. [See this book at Amazon.com]
  • The USSR Olympiad Problem Book: Selected Problems and Theorems of Elementary Mathematics.
    D.O. Shklarsky, N.N. Chentzov, and I.M. Yaglom (I. Sussman, ed.).
    3rd ed., Freeman, 1962 (?). [See this book at Amazon.com]
  • The Canadian Mathematical Olympiad 1969-1993.
    Michael Doob.
    Canadian Mathematical Society, 1993. [See this book at Amazon.com]
  • Leningrad Mathematical Olympiads 1987-1991.
    Dmitry Fomin and Alexey Kirichenko.
    MathPro Press, 1994 [See this book at Amazon.com]
  • Math Olympiad Contest Problems for Elementary and Middle Schools.
    George Lenchner.
    Glenwood, 1996. [See this book at Amazon.com]
  • The Mathematical Olympiad Handbook: An Introduction to Problem Solving Based on the First 32 British Mathematical Olympiads 1965-1996.
    Anthony David Gardiner.
    Oxford University Press, 1997. [See this book at Amazon.com]
    From the preface: ``This is unashamedly a book for beginners. Unlike most Olympiad problem books, my aim has been to convince as many people as possible that Mathematical Olympiad problem are for them and not just for some bunch of freaks.''
  • A Mathematical Mosaic: Patterns & Problem-Solving.
    Ravi Vakil.
    Brendan Kelly Publishing Inc., 1996.
    [See this book at Amazon.com]
  • Aufgaben und Lehrsätze aus der Analysis (in German).
    George Pólya and Gabor Szegö.
    Band I und II, 4th ed., Springer, 1970/1971.
    English translation:
    Problems and Theorems in Analysis.
    Springer:

Anthologies and Surveys

A delightful book, covering "old" and "new" mathematics, in 16 short well-illustrated chapters:

Don't let the title of Gower's Very Short Introduction put you off. It is very well done, even if you (think you) know mathematics well enough:

In a rare combination of history, biography and mathematics, the following books presents twelve great theorems:

A four-volume collection about and with mathematics, containing many classical essays by famous mathematicians:

  • The World of Mathematics: A Small Library of the Literature of Mathematics from A`h-mosé the Scribe to Albert Einstein.
    James R. Newman (editor).
    4 Volumes, Tempus, reprinted 1988 (original 1956). [See this book at Amazon.com]

Another, more recent, collection is:

An addictive and brilliant book about mathematics:

  • Mathematics: The Science of Patterns: The Search for Order in Life, Mind and the Universe.
    Keith Devlin.
    Scientific American Library, Freeman, 1994, Reprinted 1997. [See this book at Amazon.com]

A very accessible book that emphasizes the "Aha"-feeling when discovering (beautiful) proofs in mathematics based on varied and real mathematical problems:

A classic survey of the field of mathematics:

  • What Is Mathematics?: An Elementary Approach to Ideas and Methods.
    Richard Courant and Herbert Robbins.
    Oxford Univ. Press, 1941, 1969. (2nd ed. with Ian Stewart) [See this book at Amazon.com]

A philosophically more advanced survey is:

  • Pour l'honneur de l'esprit humain (in French).
    Jean Dieudonné
    ..., 1987.
    English translation: Mathematics: The Music of Reason. Springer, 1992. [See this book at Amazon.com]

Number Theory

The bible of number theory is:

If you are willing to fill in some gaps and want to delve into important number theory in less than 100 pages, including excercises, then go for:

Another extremely useful reference dealing with numbers is:

A somewhat excentric collection about all kinds of numbers:

A collection of problems, hints, and solutions in number theory:

  • 104 Number Theory Problems : From the Training of the USA IMO Team. Titu Andreescu, Dorin Andrica, Zuming Feng.
    Birkhäuser, 2006. [See this book at Amazon.com]

     

Algebra

Combinatorics

I am biased about the next book, because I took many (excellent) math courses from Jack van Lint. It has very broad coverage (including Pólya's Counting Theorem):

  • A Course in Combinatorics.
    J. H. van Lint and R. M. Wilson.
    Cambridge University Press; 2nd edition, 2001. [See this book at Amazon.com]

An exceptional title for an exceptional book:

An overview of theory combined with a collection of problems, hints, and solutions in combinatorics, combinatorial number theory, and combinatorial geometry:

Another collection of problems, hints, and solutions in combinatorics:

  • 102 Combinatorial Problems : From the Training of the USA IMO Team.
    Titu Andreescu, Zuming Feng.
    Birkhäuser, 2003. [See this book at Amazon.com]

     

Analysis

Geometry

  • Anschauliche Geometrie (in German).
    D. Hilbert and S. Cohn-Vossen.
    Springer, 1932.
    English translation: Geometry and the Imagination. 2nd ed.
    Chelsea Publishing Company, 1990. [See this book at Amazon.com]
  • Introduction to Geometry, 2nd ed.
    H.S.M. Coxeter.
    John Wiley & Sons, 1969. [See this book at Amazon.com]
  • Geometry Revisited.
    H.S.M. Coxeter and Samuel L. Greitzer.
    The Mathematical Association of America, 1996. [See this book at Amazon.com]
  • Challenges In Geometry: For Mathematical Olympians Past And Present.
    Christopher J. Bradley.
    Oxford University Press, 2005. [See this book at Amazon.com]
    Mostly concerned with number-theoretic and combinatoric problems based on geometric configurations

Equations and Inequalities

A light introduction to inequalities:

  • Introduction to Inequalities.
    Edwin F. Beckenbach and R. Bellman. Michael Steele.
    Mathematical Association of America, 1975.
    [See this book at Amazon.com]

An introduction to inequalities aimed at training for math contests:

  • Inequalities.
    Radmila Bulajich Manfrino, José Antonio Gómez Ortega, Rogelio Valdez Delgado.
    Instituto de Matemáticas, Universidad Nacional Autónoma de México, 2005.
    [See this book at IM UNAM]

A thoroough and more advanced introduction to inequalities:

  • The Cauchy-Schwarz Master Class: An Introduction to the Art of Inequalities.
    J. Michael Steele.
    Cambridge University Press, 2004.
    [See this book at Amazon.com]

A systematic approach to solving functional equations:

Another systematic approach to solve functional equations, based on problem from Mathematical Olympiads and other contests:

  • Functional Equations: A Problem Solving Approach.
    B. J. Venkatachala.
    Prism Books, 2002.
    ISBN 81-7286-265-2.
    [Not available at Amazon.com. Can be ordered at Eswar.com, India]

     

An overview of theory combined with a collection of problems, hints, and solutions in equations, identities, and inequalities:

Probability

  • An Introduction to Probability Theory and Its Applications.
    William Feller.
    Volume 1, 3rd ed., John Wiley & Sons, 1968. [See this book at Amazon.com]

Foundations

History of Mathematics

Mathematical Circles

  • Mathematical Circles: Russian Experience.
    Dmitry Fomin, Sergey Genkin, Ilia Itenberg.
    American Mathematical Society, 1996. [See this book at Amazon.com]
  • Out of the Labyrinth: Setting Mathematics Free.
    Robert Kaplan and Ellen Kaplan.
    Oxford University Press, 2007. [See this book at Amazon.com]
  • A Decade of the Berkeley Math Circle: The American Experience, Volume 1. =====NEW IN LIST=====
    Zvezdelina Stankova, Tom Rike.
    American Mathematical Society, 2008. [See this book at Amazon.com]
  • Circle in a Box. =====NEW IN LIST=====
    Sam Vandervelde.
    American Mathematical Society, 2009. [See this book at Amazon.com]

Other Recommendations

If you want to bring your school library up to date:

Kiran Kedlaya maintains a list of Olympiad Recommended Reading.

`Bedside' Mathematics Literature

Here follow some references to books that may not directly improve performance at the IMO, but that put mathematics and its players into a broader perspective.

Note that Martin Gardner, Ian Stewart (among others) are authors of many interesting mathematics books.

In alphabetic order of (first) author:

For the Very Young

Of interest to parents (e.g. the book shows how very young children can be taught the concepts related to quantity; this is not done by teaching them to count, but by teaching them to grasp quantity as a Gestalt): How to Teach Your Baby Math: More Gentle Revolution.
Janet Doman and Glenn Doman.
Avery Publishing Group, 1993. [See this book at Amazon.com]

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