Olympiad Books Ambitious List(1)

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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 
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PROBLEM SOLVING
The classic book about solving mathematical problems is: 
How to Solve It: A New Aspect of Mathematical Method.
G. Polya
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.
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): 
The Art and Craft of Problem Solving.
Paul Zeitz.
Excellent IMO training material: 
Problem-Solving Strategies.
Arthur Engel.
A good initial preparation for IMO-style problem solving: 
A Primer for Mathematics Competitions.
Alexander Zawaira, Gavin Hitchcock.
More training material: 
Problem Solving Through Problems.
Loren C. Larson.
From a 1988 IMO gold-medal winner, summarizing Polya's method and applying it to 26 diverse contest problems: 
Solving Mathematical Problems: A Personal Perspective
Terence Tao
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.
Based on the various (levels of) USA mathematical competitions: 
First Steps for Math Olympians: Using the American Mathematics Competitions.
J. Douglas Faires.
Approaching problem solving via puzzles and games: 
Problem Solving Through Recreational Mathematics.
Bonnie Averbach, Orin Chein.
Another general overview with many sample problems: 
Techniques of Problem Solving.
Steven G. Krantz.
Another classic on mathematical problem solving, adding a psychological perspective: 
How to Solve Mathematical Problems.
Wayne A. Wickelgren.
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COLLECTION OF MATHEMATICAL (COMPETITION) PROBLEMS
The IMO Compendium: A Collection of Problems Suggested for The International Mathematical Olympiads: 1959-2004.
Dusan Djukic, Vladimir Z. Jankovic, Ivan Matic, Nikola Petrovic.
The Art of Mathematics: Coffee Time in Memphis.
Béla Bollobás.
102 Combinatorial Problems : From the Training of the USA IMO Team.
Titu Andreescu, Zuming Feng.
103 Trigonometry Problems : From the Training of the USA IMO Team.
Titu Andreescu, Zuming Feng.
104 Number Theory Problems : From the Training of the USA IMO Team.
Titu Andreescu, Dorin Andrica, Zuming Feng.
Seeking Solutions : Discussion and Solutions of the Problems from the International Mathematical Olympiads 1988-1990.
J. C. Burns.
101 Problems in Algebra : From the Training of the USA IMO Team.
Titu Andreescu, Zuming Feng.
Introductions to Number Theory and Inequalities.
J. C. Burns.
Plane Euclidean Geometry.
A. D. Gardiner and C. J. Bradley.
Mathematical Puzzles: A Connoisseur's Collection.
Peter Winkler.
The Math Problems Notebook.
Valentin Boju, Louis Funar.
Ants, Bikes, and Clocks: Problem Solving for Undergraduates.
William Briggs.
International Mathematical Olympiads, 1955-1977.
Samuel L. Greitzer (ed.).
International Mathematical Olympiads, 1978-1985, and Forty Supplementary Problems.
Murray S. Klamkin (ed.).
International Mathematical Olympiads, 1986-1999.
Marcin E. Kuczma (ed.).
USA Mathematical Olympiads 1972-1986 Problems and Solutions.
Murray S. Klamkin (ed.).
Five Hundred Mathematical Challenges.
Edward J. Barbeau, Murray S. Klamkin, William O. J. Moser.
From Erdos to Kiev : Problems of Olympiad Caliber.
Ross Honsberger.
In Pólya's Footsteps: Miscellaneous Problems and Essays.
Ross Honsberger.
Mathematical Miniatures.
Svetsoslav Savchev and Titu Andreescu.
Problems from Murray Klamkin.
Andy Liu, Bruce Shawyer (Eds.).
Aha! Solutions.
Martin Erickson.
Mathematical Olympiad Challenges.
by Titu Andreescu and Razvan Gelca.
Mathematical Olympiad Treasures.
by Titu Andreescu and Bogden Eneescu.
Winning Solutions.
Edward Lozansky and Cecil Rousseau.
Colorado Mathematical Olympiad: The First Ten Years and Further Explorations.
Alexander Soifer.
New Mexico Mathematics Contest Problem Book.
Liong-shin Hahn.
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.).
The Canadian Mathematical Olympiad 1969-1993.
Michael Doob.
Leningrad Mathematical Olympiads 1987-1991.
Dmitry Fomin and Alexey Kirichenko.
Math Olympiad Contest Problems for Elementary and Middle Schools.
George Lenchner.
The Mathematical Olympiad Handbook: An Introduction to Problem Solving Based on the First 32 British Mathematical Olympiads 1965-1996.
Anthony David Gardiner.
A Mathematical Mosaic: Patterns & Problem-Solving.
Ravi Vakil.
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ANTHOLOGIES & SURVEYS
A delightful book, covering "old" and "new" mathematics, in 16 short well-illustrated chapters: 
1089 and All That: A Journey into Mathematics.
David Acheson.
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: 
Mathematics: A Very Short Introduction.
Timothy Gowers.
In a rare combination of history, biography and mathematics, the following books presents twelve great theorems: 
Journey through Genius: The Great Theorems of Mathematics.
William Dunham.
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).
Another, more recent, collection is: 
Mathematics: People, Problems, Results.
D. M. Campbell and J. C. Higgins.
An addictive and brilliant book about mathematics: 
Mathematics: The Science of Patterns: The Search for Order in Life, Mind and the Universe.
Keith Devlin.
 very accessible book that emphasizes the "Aha"-feeling when discovering (beautiful) proofs in mathematics based on varied and real mathematical problems: 
The Moment of Proof : Mathematical Epiphanies.
Donald C. Benson.
A classic survey of the field of mathematics: 
What Is Mathematics?: An Elementary Approach to Ideas and Methods.
Richard Courant and Herbert Robbins.
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NUMBER THEORY
The bible of number theory is: 
An Introduction to the Theory of Numbers. 5th ed.
G.H. Hardy and E.M. Wright.
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: 
A Concise Introduction to the Thoory of Numbers.
Alan Baker.
Another extremely useful reference dealing with numbers is: 
The Encyclopedia of Integer Sequences.
N.J.A. Sloane and Simon Plouffe.
A somewhat excentric collection about all kinds of numbers: 
The Books of Numbers.
John Horton Conway, Richard K. Guy.
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
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ALGEBRA
A Survey of Modern Algebra.
Saunders MacLane and Garrett D. Birkhoff.
Polynomials.
Edward J. Barbireau.
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COMBINATORICS
A Course in Combinatorics.
J. H. van Lint and R. M. Wilson.
An exceptional title for an exceptional book: 
Generatingfunctionology.
Herbert Wilf
An overview of theory combined with a collection of problems, hints, and solutions in combinatorics, combinatorial number theory, and combinatorial geometry: 
Counting and Configurations.
Jiri Herman, Radan Kucera, and Jaromir Simsa.
Another collection of problems, hints, and solutions in combinatorics: 
102 Combinatorial Problems : From the Training of the USA IMO Team.
Titu Andreescu, Zuming Feng.
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ANALYSIS
A Course of Pure Mathematics. 10th ed.
G.H. Hardy.
Calculus. 3rd ed.
Michael Spivak.
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GEOMETRY
Introduction to Geometry, 2nd ed.
H.S.M. Coxeter.
Geometry Revisited.
H.S.M. Coxeter and Samuel L. Greitzer.
Challenges In Geometry: For Mathematical Olympians Past And Present.
Christopher J. Bradley.
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EQUATIONS & INEQUALITIES
 light introduction to inequalities: 
Introduction to Inequalities.
Edwin F. Beckenbach and R. Bellman. Michael Steele.
An introduction to inequalities aimed at training for math contests: 
Inequalities.
Radmila Bulajich Manfrino, José Antonio Gómez Ortega, Rogelio Valdez Delgado.
A thoroough and more advanced introduction to inequalities: 
The Cauchy-Schwarz Master Class: An Introduction to the Art of Inequalities.
J. Michael Steele.
A systematic approach to solving functional equations: 
Functional Equations and How to Solve Them.
Christopher G. Small.
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.
An overview of theory combined with a collection of problems, hints, and solutions in equations, identities, and inequalities: 
Equations and Inequalities.
Jiri Herman, Radan Kucera, and Karl Dilcher.
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PROBABILITY
An Introduction to Probability Theory and Its Applications.
William Feller.
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FOUNDATIONS
Naive Set Theory.
Paul R. Halmos.
Gödel's Proof.
Ernest Nagel and James R. Newman.
The Foundations of Mathematics.
Ian Stewart and David Tall.
Conceptual Mathematics: A First Introduction to Categories.
F. William Lawvere and Stephen H. Schanuel.
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HISTORY OF MATHEMATICS
Men of Mathematics.
Eric Temple Bell.
Mathematical Thought from Ancient to Modern Times. 3 volumes.
Morris Kline.
A Concise History of Mathematics.
Dirk J. Struik
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MATHEMATICAL CIRCLES
Mathematical Circles: Russian Experience.
Dmitry Fomin, Sergey Genkin, Ilia Itenberg.
Out of the Labyrinth: Setting Mathematics Free.
Robert Kaplan and Ellen Kaplan.
A Decade of the Berkeley Math Circle: The American Experience, Volume 1.
Zvezdelina Stankova, Tom Rike.
Circle in a Box.
Sam Vandervelde.
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