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Honsberger revisited:mathematical gems polished

出版日期
2011
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PDF
書籍分類
學科分類
ISBN
9789860311686

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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.”
  • Introduction
  • Preface
  • 1 Mathematical Delights
    • 1.1 Triangles in Orthogonal Position
    • 1.2 Pan Balance
    • 1.3 Schoch 3
    • 1.4 A Nice Problemin Probability
    • 1.5 Three Proofs of the Heron Formula
    • 1.6 Incenter
    • 1.7 On MedianAltitude 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 Pólya’s Footsteps
    • 2.1 Curious Squares
    • 2.2 A Problem from 15th Russian Olympiad
    • 2.3 MaximumWithout 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 AWell-Known Figure
    • 4.4 Rangers withWalkie-Talkie
    • 4.5 A Piston Rod
    • 4.6 The Schwab-SchoenbergMean
    • 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 Erdös to Kiev
    • 5.1 The Sum of Consecutive Positive Integers
    • 5.2 A Problemin 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 Kürschák 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 Olympiad1985
  • 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
      • C.2.3 The Simson Line
    • C.3 Trigonometry
      • C.3.1 Trigonometric Series
      • C.3.2 The Laws of Sines and Cosines
    • C.4 Conics
      • C.4.1 Parametric Equations
      • C.4.2 Tangents
    • C.5 The Jensen Inequality

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