Introduction

What is it about math problems that makes them so addictive?

When I get going on a problem, I’d rather stay at it than eat!

Liong-shin Hahn and I are kindred spirits who have lived like this for the last fifty years. As you might expect, Liong-shin has become very adept at solving problems, and in this volume he has collected his treatments of some hundred problems that caught his eye in my books. If you would like a sample of his ingenuity, take a look at Exercise 49 (pages 149 and 215) or his solution to Exercise 8 (pages 138 and 177); and wait till you see what Problem 2.1 (page 29) conjured up in his mind!

While this volume might be used in the training of young scholars to write mathematics contests, it is more than that. This is a book for everyone who delights in the richness, beauty, and excitement of the wonderful ideas that abide in the realms of elementary mathematics. I feel it is only fair to caution you that this book can lead to a deeper appreciation and love of mathematics.

Without further ado, then, let us turn over the stage to this remarkable man—Liong-shin Hahn.



By Ross Honsberger.

Introduction

What is it about math problems that makes them so addictive?

When I get going on a problem, I’d rather stay at it than eat!

Liong-shin Hahn and I are kindred spirits who have lived like this for the last fifty years. As you might expect, Liong-shin has become very adept at solving problems, and in this volume he has collected his treatments of some hundred problems that caught his eye in my books. If you would like a sample of his ingenuity, take a look at Exercise 49 (pages 14...

Preface

Before my retirement, I was responsible for composing the problems for the New Mexico Mathematics Contest1. Whenever I needed an inspiration then, I always turned to the wonderful books by Professor Ross Honsberger. I knew his books were a treasure trove of interesting problems with brilliant solutions. So it was only natural that when I retired and wanted to prevent dementia, I chose problems from his books for my “Problem-of-the-Day” activity. I would not peek at the solution unless I had solved the problem myself, or could not come up with a fresh approach to tackle the problem for at least 72 hours. The problem-of-the-day activity gave me a daily drama. Some days, I was delighted to have found nice solutions, but some other days I was disappointed that my “brilliant” solutions turned out to be essentially the same as those presented in his books, or worse, not so brilliant compared to the published ones.

Time and again, Professor Honsberger encouraged me to publish my solutions. This book is the consequence. I can only claim I found these solutions myself. But as it is irrelevant for my problem-of-the-day activity, no effort is made to check whether they are new. Naturally, my solutions that have already appeared in Professor Honsberger's books are excluded. On the other hand, some solutions that are not so elegant are included, in the hope that they still have some merits. By the way, my original plan was to include also Episodes in Nineteenth and Twentieth Century Euclidean Geometry, which is another very rich source for exploration. However, my manuscript is already over 300 pages, and the inclusion would make this one too lopsided toward geometry. Furthermore, after five years on this book project, I am eager to move on to the next phase of my life.

Each chapter corresponds to Professor Honsberger's book with the same title. However, the order of the chapters is random, and so they can be read in any order. Yet, in Exercises (Appendix A) and Solutions (Appendix B), I preserve the order in Professor Honsberger's books for easy reference. The source of each problem is identified by a single or a pair of number(s) in brackets. For example, because Chapter 2 corresponds to In P´olya's Footsteps, so a problem taken from page 67 of In P´olya's Footsteps is indicated by [67] in Chapter 2, while this same problem is referred to in other chapters by [2:67]. None of the problems in Exercises (Appendix A)1See my book, New Mexico Mathematics Contest Problem Book (University of New Mexico Press, 2005).popped out of thin air. If they have appeared in Professor Honsberger's books, then alternate solutions can be found in Solutions (Appendix B). Others are byproducts of my solutions. Therefore, all have their origins, directly or indirectly, in Professor Honsberger's works. Appendix C is designed to provide sufficient background for the readers. It contains my favorite “tools of the trade”.

I am an unabashed admirer of the late Professor George P´olya, mainly for the elegance of his mathematics, but also for his teaching and problem-solving methods, not to mention his devotion to mathematics education.

I keep on telling students whenever I have a chance: “If you find a book by Professor P´olya, buy it and read it. You will be happy you did.” His books are invaluable for anyone in mathematics, both in teaching and in research. I only wish I had a chance to hear his comments on my solutions. I am sure many readers can detect his influence on me.

Although Professor Honsberger's books are not a necessary background; i.e., this book can be read independently, I am sure, by parallel reading, the reader will be all the more entertained. And I certainly hope that readers who enjoy his books will also enjoy mine. At the minimum, I hope I have some success in conveying the joy of problem solving.

It is a pleasure to express my heartfelt appreciation to Professor Honsberger for his friendship over the decades, and his encouragement throughout this book project. I can never thank him enough for his very meticulous reading of the manuscript and generous help in improvement of the presentation, not to mention his endorsement in the Introduction.

L.-s. H.

February 2008

Postscript. It is a pleasure to express my deep appreciation to Dr. Luke Cheng-chung Yu (neonatology and pediatric cardiology, board certified) and my son, Shin-Yi, for their help in solving the computer problems for me. Without their help, I don’t know how long the publication of this book would have been delayed. Even though the manuscript was completed in February 2008, it was submitted to the National Taiwan University Press three years later. Knowing the book will be published within one year was a happy surprise for the author. Now the fortunate result is before you.

Preface

Before my retirement, I was responsible for composing the problems for the New Mexico Mathematics Contest1. Whenever I needed an inspiration then, I always turned to the wonderful books by Professor Ross Honsberger. I knew his books were a treasure trove of interesting problems with brilliant solutions. So it was only natural that when I retired and wanted to prevent dementia, I chose problems from his books for my “Problem-of-the-Day” activity. I would not peek at the solution unless...

Contents
Introduction vii
Preface viii

1 Mathematical Delights 1
1.1 Triangles in Orthogonal Position 1
1.2 Pan Balance 6
1.3 Schoch 3 7
1.4 A Nice Problem in Probability 10
1.5 Three Proofs of the Heron Formula 13
1.6 Incenter 18
1.7 On Median, Altitude and Angle Bisector 19
1.8 A Geometry Problem from Quantum 23
1.9 Monochromatic Triangle 25
1.10 Sum of the Greatest Odd Divisors 26
1.11 Prime Numbers of the Form m2k + mknk + n2k 27

2 In P olya's Footsteps 29
2.1 Curious Squares 29
2.2 A Problem from 15th Russian Olympiad 32
2.3 Maximum Without Calculus 34
2.4 Cocyclic Points 36
2.5 Reconstruction of the Original Triangle 37
2.6 The Sums of the Powers 38
2.7 A Problem from Crux Mathematicorum 46
2.8 A Puzzle 47
2.9 Pedal Triangle with Preasigned Shape 47
2.10 An Intriguing Geometry Problem 49

3 Mathematical Chestnuts from Around the World 52
3.1 Three Similar Triangles Sharing a Vertex 52
3.2 The Simson Line in Disguise 55
3.3 Circle through Points 56
3.4 Zigzag 57
3.5 Cevians 60
3.6 Integers of a Particular Type Divisible by 2n 61
3.7 Quadrangles with Perpendicular Diagonals 61

4 Mathematical Diamonds 64
4.1 Orthic Triangle 64
4.2 Quartering a Quadrangle 66
4.3 A Well-Known Figure 67
4.4 Rangers with Walkie-Talkie 71
4.5 A Piston Rod 73
4.6 The Schwab-Schoenberg Mean 75
4.7 Construction of an Isosceles Triangle 79
4.8 The Conjugate Orthocenter 82
4.9 A Remarkable Pair 85
4.10 Calculus? 89
4.11 A Problem from the 1980 Tournament of Towns 91

5 From Erdos to Kiev 96
5.1 The Sum of Consecutive Positive Integers 96
5.2 A Problem in Graph Theory 98
5.3 A Triangle with its Euler Line Parallel to a Side 99
5.4 A “Pythagorean” Triple 102
5.5 A Geometry Problem from the K¨ursch´ak Competition 104
5.6 A Lovely Geometric Construction 109
5.7 A Problem from the 1987 Austrian Olympiad 112
5.8 Another Problem from the 1987 Austrian Olympiad 116
5.9 An Unexpected Property of Triangles 119
5.10 Products of Consecutive Integers 125
5.11 A Problem from the Second Balkan Olympiad, 1985 129

A Exercises 136

B Solutions 166

C Useful Theorems 288
C.1 Triangles 288
C.1.1 Complex Plane 288
C.1.2 Corollaries 290
C.1.3 Equilateral Triangles 291
C.1.4 Theorems of Ceva and Menelaus 291
C.2 Circles 294
C.2.1 Subtended Angles 294
C.2.2 The Power Theorem 297

Contents
Introduction vii
Preface viii

1 Mathematical Delights 1
1.1 Triangles in Orthogonal Position 1
1.2 Pan Balance 6
1.3 Schoch 3 7
1.4 A Nice Problem in Probability 10
1.5 Three Proofs of the Heron Formula 13
1.6 Incenter 18
1.7 On Median, Altitude and Angle Bisector 19
1.8 A Geometry Problem from Quantum 23
1.9 Monochromatic Triangle 25
1.10 Sum of the Greatest Odd Divisors 26
1.11 Prime Numbers of the Form m2k + mknk + n2k 27

2 In P olya's Footst...