Mathematical Olympiad Challenges

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Springer Science & Business Media, 10 janv. 2001 - 260 pages
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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.

 

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Table des matières

11 A Property of Equilateral Triangles
4
12 Cyclic Quadrilaterals
6
13 Power of a Point
9
14 Dissections of Polygonal Surfaces
13
15 Regular Polygons
16
16 Geometric Constructions and Transformations
20
17 Problems with Physical Flavor
22
18 Tetrahedra Inscribed in Parallelepipeds
24
13 Power of a Point
99
14 Dissections of Polygonal Surfaces
106
15 Regular Polygons
114
16 Geometric Constructions and Transformations
124
17 Problems with Physical Flavor
129
18 Tetrahedra Inscribed in Parallelepipeds
136
19 Telescopic Sums and Products in Trigonometry
140
110 Trigonometric Substitutions
145

19 Telescopic Sums and Products in Trigonometry
26
110 Trigonometric Substitutions
29
Algebra and Analysis
33
21 No Square is Negative
34
22 Look at the Endpoints
36
23 Telescopic Sums and Products in Algebra
38
24 On an Algebraic Identity
41
25 Systems of Equations
43
26 Periodicity
47
27 The Abel Summation Formula
50
28 𝑥 + 1𝑥
52
29 Matrices
54
210 The Mean Value Theorem
55
Number Theory and Combinatorics
59
31 Arrange in Order
60
32 Squares and Cubes
62
33 Repunits
64
34 Digits of Numbers
66
35 Residues
69
36 Diophantine Equations with the Unknowns as Exponents
72
37 Numerical Functions
74
38 Invariants
77
39 Pell Equations
80
310 Prime Numbers and Binomial Coefficients
84
Solutions
87
Geometry and Trigonometry
89
11 A Property of Equilateral Triangles
90
12 Cyclic Quadrilaterals
93
Algebra and Analysis
151
21 No Square is Negative
152
22 Look at the Endpoints
156
23 Telescopic Sums and Products in Algebra
159
24 On an Algebraic Identity
164
25 Systems of Equations
166
26 Periodicity
172
27 The Abel Summation Formula
176
28 𝑥 + l𝑥
183
29 Matrices
188
210 The Mean Value Theorem
191
Number Theory and Combinatorics
197
31 Arrange in Order
198
32 Squares and Cubes
201
33 Repunits
206
34 Digits of Numbers
209
35 Residues
216
36 Diophantine Equations with Unknowns as Exponents
221
37 Numerical Functions
226
38 Invariants
233
39 Pell Equations
237
310 Prime Numbers and Binomial Coefficients
244
Definitions and Notation
251
AI Glossary of Terms
252
A2 Glossary of Notation
257
About the Authors
259
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À propos de l'auteur (2001)

Titu Andreescu received his Ph.D. from the West University of Timisoara, Romania. The topic of his dissertation was "Research on Diophantine Analysis and Applications." Professor Andreescu currently teaches at The University of Texas at Dallas. He is past chairman of the USA Mathematical Olympiad, served as director of the MAA American Mathematics Competitions (1998a "2003), coach of the USA International Mathematical Olympiad Team (IMO) for 10 years (1993a "2002), director of the Mathematical Olympiad Summer Program (1995a "2002), and leader of the USA IMO Team (1995a "2002). In 2002 Titu was elected member of the IMO Advisory Board, the governing body of the world's most prestigious mathematics competition. Titu co-founded in 2006 and continues as director of the AwesomeMath Summer Program (AMSP). He received the Edyth May Sliffe Award for Distinguished High School Mathematics Teaching from the MAA in 1994 and a "Certificate of Appreciation" from the president of the MAA in 1995 for his outstanding service as coach of the Mathematical Olympiad Summer Program in preparing the US team for its perfect performance in Hong Kong at the 1994 IMO. Titua (TM)s contributions to numerous textbooks and problem books are recognized worldwide.

Dorin Andrica received his Ph.D. in 1992 from "Babes-Bolyaia University in Cluj-Napoca, Romania; his thesis treated critical points and applications to the geometry of differentiable submanifolds. Professor Andrica has been chairman of the Department of Geometry at "Babes-Bolyai" since 1995. He has written and contributed to numerous mathematics textbooks, problem books, articles and scientific papers at various levels. He is an invited lecturer atuniversity conferences around the world: Austria, Bulgaria, Czech Republic, Egypt, France, Germany, Greece, Italy, the Netherlands, Portugal, Serbia, Turkey, and the USA. Dorin is a member of the Romanian Committee for the Mathematics Olympiad and is a member on the editorial boards of several international journals. Also, he is well known for his conjecture about consecutive primes called "Andrica's Conjecture." He has been a regular faculty member at the Canadaa "USA Mathcamps between 2001a "2005 and at the AwesomeMath Summer Program (AMSP) since 2006.

Zuming Feng received his Ph.D. from Johns Hopkins University with emphasis on Algebraic Number Theory and Elliptic Curves. He teaches at Phillips Exeter Academy. Zuming also served as a coach of the USA IMO team (1997-2006), was the deputy leader of the USA IMO Team (2000-2002), and an assistant director of the USA Mathematical Olympiad Summer Program (1999-2002). He has been a member of the USA Mathematical Olympiad Committee since 1999, and has been the leader of the USA IMO team and the academic director of the USA Mathematical Olympiad Summer Program since 2003. Zuming is also co-founder and academic director of the AwesomeMath Summer Program (AMSP) since 2006. He received the Edyth May Sliffe Award for Distinguished High School Mathematics Teaching from the MAA in 1996 and 2002.

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