Honsberger Revisited：Mathematical Gems Polished

　　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.”

作者簡介

Liong-shin Hahn

　　Liong-shin Hahn was born into a family of physicians in Tainan, Taiwan.  He calls himself the black sheep of the family, because, like his father, Shyr-Chyuan Hahn, M.D., Ph.D., all five of his brothers became physicians.  After graduating from Tainan First Senior Middle School and the National Taiwan University, he attended Stanford University and obtained his Ph.D. there under Professor Karel deLeeuw.  He spent most of his career at the University of New Mexico, and while away from that institution, he held visiting positions at the University of Washington (Seattle), the National Taiwan University, the University of Tokyo, Sophia University (Tokyo) and the International Christian University (Tokyo).   As director of the mathematics contest sponsored by the University of New Mexico, he consulted frequently the superb books by Professor Ross Honsberger that seeded the birth of this book.  He authored Complex Numbers and Geometry (Mathematicial Association of  Americia, 1994), New Mexico Mathematics Contest Problem Book (University of New Mexico Press, 2005), and co-authored with Bernard Epstein Classical Complex Analysis (Jones and Bartlett, 1996).  He was awarded the Citation for Public Service from the American Mathematical Society in 1998.  His marriage to Hwei-Shien Lee (yet another M.D.) yielded three sons and seven grandchildren.

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