Mathematical Olympiad Challenges

Couverture
Springer Science & Business Media, 26 avr. 2000 - 260 pages

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
xiv
13 Power of a Point
1
14 Dissections of Polygonal Surfaces
5
15 Regular Polygons
8
16 Geometric Constructions and Transformations
12
17 Problems with Physical Flavor
14
18 Tetrahedra Inscribed in Parallelepipeds
16
19 Telescopic Sums and Products in Trigonometry
18
13 Power of a Point
91
14 Dissections of Polygonal Surfaces
98
15 Regular Polygons
106
16 Geometric Constructions and Transformations
116
17 Problems with Physical Flavor
121
18 Tetrahedra Inscribed in Parallelepipeds
128
19 Telescopic Sums and Products in Trigonometry
132
110 Trigonometric Substitutions
137

110 Trigonometric Substitutions
21
Algebra and Analysis
25
21 No Square is Negative
26
22 Look at the Endpoints
28
23 Telescopic Sums and Products in Algebra
30
24 On an Algebraic Identity
33
25 Systems of Equations
35
26 Periodicity
39
27 The Abel Summation Formula
42
28 x + 1x
44
29 Matrices
46
210 The Mean Value Theorem
47
Number Theory and Combinatorics
51
31 Arrange in Order
52
32 Squares and Cubes
54
33 Repunits
56
34 Digits of Numbers
58
35 Residues
61
36 Diophantine Equations with the Unknowns as Exponents
64
37 Numerical Functions
66
38 Invariants
69
39 Pell Equations
72
310 Prime Numbers and Binomial Coefficients
76
Solutions
79
Geometry and Trigonometry
81
11 A Property of Equilateral Triangles
82
12 Cyclic Quadrilaterals
85
Algebra and Analysis
143
21 No Square is Negative
144
22 Look at the Endpoints
148
23 Telescopic Sums and Products in Algebra
151
24 On an Algebraic Identity
156
25 Systems of Equations
158
26 Periodicity
164
27 The Abel Summation Formula
168
28 x + 1x
175
29 Matrices
180
210 The Mean Value Theorem
183
Number Theory and Combinatorics
189
31 Arrange in Order
190
32 Squares and Cubes
193
33 Repunits
198
34 Digits of Numbers
201
35 Residues
208
36 Diophantine Equations with Unknowns as Exponents
213
37 Numerical Functions
218
38 Invariants
225
39 Pell Equations
229
310 Prime Numbers and Binomial Coefficients
236
Appendix A Definitions and Notation
243
A1 Glossary of Terms
244
A2 Glossary of Notation
249
About the Authors
251
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