The author’s purpose is to share the thrills and excitement of ingenious solutions to intriguing elementary problems that he has had the good fortune to have conceived in the pursuit of his passion over many years. His satisfaction lies in the beauty of these gems, not in the incidental fact that they happen to be his own work. A wonderful solution is a glorious thing, whoever might have thought of it, and the author has worked diligently to make easy reading of the joy and delights of his often hard-won success.

As Director, responsible for composing the problems for the New Mexico Mathematics Contest before his retirement, the author consulted the wonderful books by Professor Ross Honsberger whenever he needed an inspiration. As a result, the New Mexico Mathematics Contest rose to national prominence and the author received the “Citation for Public Service” from the American Mathematical Society in 1998. In this volume he collected his treatments of over a hundred problems from the treasure trove of Professor Honsberger.

Perhaps it is best to quote Professor Honsberger, “This is a book for everyone who delights in the richness, beauty, and excitement of the wonderful ideas that abide in the realm 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.”

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.

Contents

Introduction

Preface

1 Mathematical Delights

1.1 Triangles in Orthogonal Position

1.2 Pan Balance

1.3 Schoch 3

1.4 A Nice Problem in Probability

1.5 Three Proofs of the Heron Formula

1.6 Incenter

1.7 On Median, Altitude and Angle Bisector

1.8 A Geometry Problem from Quantum

1.9 Monochromatic Triangle

1.10 Sum of the Greatest Odd Divisors

1.11 Prime Numbers of the Form m2k + mknk + n2k

2 In P olya’s Footsteps

2.1 Curious Squares

2.2 A Problem from 15th Russian Olympiad

2.3 Maximum Without Calculus

2.4 Cocyclic Points

2.5 Reconstruction of the Original Triangle

2.6 The Sums of the Powers

2.7 A Problem from Crux Mathematicorum

2.8 A Puzzle

2.9 Pedal Triangle with Preasigned Shape

2.10 An Intriguing Geometry Problem

3 Mathematical Chestnuts from Around the World

3.1 Three Similar Triangles Sharing a Vertex

3.2 The Simson Line in Disguise

3.3 Circle through Points

3.4 Zigzag

3.5 Cevians

3.6 Integers of a Particular Type Divisible by 2n

3.7 Quadrangles with Perpendicular Diagonals

4 Mathematical Diamonds

4.1 Orthic Triangle

4.2 Quartering a Quadrangle

4.3 A Well-Known Figure

4.4 Rangers with Walkie-Talkie

4.5 A Piston Rod

4.6 The Schwab-Schoenberg Mean

4.7 Construction of an Isosceles Triangle

4.8 The Conjugate Orthocenter

4.9 A Remarkable Pair

4.10 Calculus?

4.11 A Problem from the 1980 Tournament of Towns

5 From Erd�os to Kiev

5.1 The Sum of Consecutive Positive Integers

5.2 A Problem in Graph Theory

5.3 A Triangle with its Euler Line Parallel to a Side

5.4 A “Pythagorean” Triple

5.5 A Geometry Problem from the K‥ursch′ak Competition

5.6 A Lovely Geometric Construction

5.7 A Problem from the 1987 Austrian Olympiad

5.8 Another Problem from the 1987 Austrian Olympiad

5.9 An Unexpected Property of Triangles

5.10 Products of Consecutive Integers

5.11 A Problem from the Second Balkan Olympiad, 1985

A Exercises

B Solutions

C Useful Theorems

C.1 Triangles

C.1.1 Complex Plane

C.1.2 Corollaries

C.1.3 Equilateral Triangles

C.1.4 Theorems of Ceva and Menelaus

C.2 Circles

C.2.1 Subtended Angles

C.2.2 The Power Theorem