Problem-Solving Strategies

✍ Scribed by Arthur Engel

• Publisher: Springer
• Year: 1997
• Tongue: English
• Leaves: 415
• Series: Problem Books in Mathematics
• Edition: Corrected
• Category: Library

✦ Synopsis

A unique collection of competition problems from over twenty major national and international mathematical competitions for high school students. Written for trainers and participants of contests of all levels up to the highest level, this will appeal to high school teachers conducting a mathematics club who need a range of simple to complex problems and to those instructors wishing to pose a "problem of the week", thus bringing a creative atmosphere into the classrooms. Equally, this is a must-have for individuals interested in solving difficult and challenging problems. Each chapter starts with typical examples illustrating the central concepts and is followed by a number of carefully selected problems and their solutions. Most of the solutions are complete, but some merely point to the road leading to the final solution. In addition to being a valuable resource of mathematical problems and solution strategies, this is the most complete training book on the market.

✦ Table of Contents

Cover......Page 1
Series: Problem Books in Mathematics......Page 2
Problem-Solving Strategies......Page 4
Copyright......Page 5
Preface......Page 6
Contents......Page 8
Abbreviations and Notations......Page 10
1. The Invariance Principle......Page 12
Problems......Page 19
Solutions......Page 25
2. Coloring Proofs......Page 36
Problems......Page 37
Solutions......Page 39
3. The Extremal Principle......Page 50
Problems......Page 58
Solutions......Page 61
4. The Box Principle......Page 70
Ramsey Numbers, Sum-Free Sets, and a Theorem of I. Schur......Page 74
Problems......Page 79
Solutions......Page 84
5. Enumerative Combinatorics......Page 96
Problems......Page 111
Solutions......Page 114
6. Number Theory......Page 128
Divisibility......Page 132
Problems......Page 142
Solutions......Page 148
Means......Page 172
Strategies for Proving Inequalities......Page 189
Problems......Page 190
Solutions......Page 197
8. The Induction Principle......Page 216
Problems......Page 218
Solutions......Page 220
9. Sequences......Page 232
Problems......Page 236
Solutions......Page 240
10. Polynomials......Page 256
Problems......Page 265
Solutions......Page 269
11. Functional Equations......Page 282
Problems......Page 288
Solutions......Page 291
12.1.1 Affine Geometry......Page 300
12.1.2 Scalar or Dot Product......Page 303
12.1.3 Complex Numbers......Page 305
Problems......Page 309
Solutions......Page 312
13. Games......Page 372
Problems......Page 373
Solutions......Page 376
14.1 Graph Theory......Page 384
14.2 Infinite Descent......Page 385
14.3 Working Backwards......Page 388
14.4 Conjugate Numbers......Page 389
Problems......Page 390
14.5 Equations, Functions, and Iterations......Page 391
Problems......Page 392
14.6 Integer Functions......Page 393
Problems......Page 394
Solutions......Page 395
References......Page 408
Index......Page 412