Mathematical Olympiad Challenges is a rich collection of problems put together by two experienced and well-known professors and coaches of the U.S. International Mathematical Olympiad Team. Hundreds of beautiful, challenging, and instructive problems from algebra, geometry, trigonometry, combinatorics, and number theory were selected from numerous mathematical competitions and journals. An important feature of the work is the comprehensive background material provided with each grouping of problems.
The problems are clustered by topic into self-contained sections with solutions provided separately. All sections start with an essay discussing basic facts and one or two representative examples. A list of carefully chosen problems follows and the reader is invited to take them on. Additionally, historical insights and asides are presented to stimulate further inquiry. The emphasis throughout is on encouraging readers to move away from routine exercises and memorized algorithms toward creative solutions to open-ended problems.
Aimed at motivated high school and beginning college students and instructors, this work can be used as a text for advanced problem- solving courses, for self-study, or as a resource for teachers and students training for mathematical competitions and for teacher professional development, seminars, and workshops.
Rezensionen / Stimmen
"The authors are experienced problem solvers and coaches of mathematics teams. This expertise shows through and the result is a volume that would be a welcome addition to any mathematician's bookshelf." -MAA Online "This [book] is...much more than just another collection of interesting, challenging problems, but is instead organized specifically for learning. The book expertly weaves together related problems, so that insights gradually become techniques, tricks slowly become methods, and methods eventually evolve into mastery... The book is aimed at motivated high school and beginning college students and instructors. It can be used as a text for advanced problem-solving courses, for self-study, or as a resource for teachers and students training for mathematical competitions, and for teacher professional development, seminars, and workshops. I strongly recommend this book for anyone interested in creative problem-solving in mathematics... It has already taken up a prized position in my personal library, and is bound to provide me with many hours of intellectual pleasure." -The Mathematical Gazette
Auflage
1st ed. 2000. Corr. 2nd printing
Sprache
Verlagsort
Zielgruppe
Für höhere Schule und Studium
Research
Editions-Typ
Produkt-Hinweis
Illustrationen
80 black & white illustrations, 80 black & white line drawings
Maße
Höhe: 27.9 cm
Breite: 21.6 cm
Dicke: 15 mm
Gewicht
ISBN-13
978-0-8176-4155-9 (9780817641559)
DOI
10.1007/978-1-4612-2138-8
Schweitzer Klassifikation
I Problems.- 1 Geometry and Trigonometry.- 1.1 A Property of Equilateral Triangles.- 1.2 Cyclic Quadrilaterals.- 1.3 Power of a Point.- 1.4 Dissections of Polygonal Surfaces.- 1.5 Regular Polygons.- 1.6 Geometric Constructions and Transformations.- 1.7 Problems with Physical Flavor.- 1.8 Tetrahedra Inscribed in Parallelepipeds.- 1.9 Telescopic Sums and Products in Trigonometry.- 1.10 Trigonometric Substitutions.- 2 Algebra and Analysis.- 2.1 No Square is Negative.- 2.2 Look at the Endpoints.- 2.3 Telescopic Sums and Products in Algebra.- 2.4 On an Algebraic Identity.- 2.5 Systems of Equations.- 2.6 Periodicity.- 2.7 The Abel Summation Formula.- 2.8 x + 1/x.- 2.9 Matrices.- 2.10 The Mean Value Theorem.- 3 Number Theory and Combinatorics.- 3.1 Arrange in Order.- 3.2 Squares and Cubes.- 3.3 Repunits.- 3.4 Digits of Numbers.- 3.5 Residues.- 3.6 Equations with Unknowns as Exponents.- 3.7 Numerical Functions.- 3.8 Invariants.- 3.9 Pell Equations.- 3.10 Prime Numbers and Binomial Coefficients.- II Solutions.- 1 Geometry and Trigonometry.- 1.1 A Property of Equilateral Triangles.- 1.2 Cyclic Quadrilaterals.- 1.3 Power of a Point.- 1.4 Dissections of Polygonal Surfaces.- 1.5 Regular Polygons.- 1.6 Geometric Constructions and Transformations.- 1.7 Problems with Physical Flavor.- 1.8 Tetrahedra Inscribed in Parallelepipeds.- 1.9 Telescopic Sums and Products in Trigonometry.- 1.10 Trigonometric Substitutions.- 2 Algebra and Analysis.- 2.1 No Square is Negative.- 2.2 Look at the Endpoints.- 2.3 Telescopic Sums and Products in Algebra.- 2.4 On an Algebraic Identity.- 2.5 Systems of Equations.- 2.6 Periodicity.- 2.7 The Abel Summation Formula.- 2.8 x + l/x.- 2.9 Matrices.- 2.10 The Mean Value Theorem.- 3 Number Theory and Combinatorics.- 3.1 Arrange in Order.- 3.2 Squares and Cubes.- 3.3 Repunits.- 3.4 Digits of Numbers.- 3.5 Residues.- 3.6 Equations with Unknowns as Exponents.- 3.7 Numerical Functions.- 3.8 Invariants.- 3.9 Pell Equations.- 3.10 Prime Numbers and Binomial Coefficients.- A Appendix A: Definitions and Notation.- A.1 Glossary of Terms.- A.2 Glossary of Notation.- About the Authors.
