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Mathematical Morsels Ross Honsberger : AMS / MAA DOLCIANI MATHEMATICAL EXPOSITIONS VOL 10 More Mathematical Morsels Ross Honsberger :"10RE MATHEMATICAL MORSELS ROSS HONSBERGER THE DOLeIANI MATHEMATICAL EXPOSITIONS Published by THE MATHEMATICAL ASSOCIATION OF AMERICA Committee on Publications DONALD ALBERS, Chairman Dolciani Mathematical Expositions Editorial Board JOE ~ BUHLER, Chairman JOHN H. EWING JOHNR.LUX WILLIAM WATERHOUSE Dolciani Mathematical Expositions series design ROBERT ISHI + DESIGN TEXmacros RON WHITNEY 10.1090/dol/010 The Dolciani Mathematical Expositions NUMBER TEN MORE MATHEMATICAL MORSELS ROSS HONSBERGER University ofWaterloo Published and Distributed by THE MATHEMATICAL ASSOCIATION OF AMERICA @1991 by The Mathematical Association ofAmerica (Incorporated) Library of Congress Catalog Card Number 90-70792 Complete Set ISBN 0-88385-300-0 Vol. 10 ISBN 0-88385-314-0 Printed in the United States ofAmerica Current printing (last digit): 10 9 8 7 6 5 4 3 2 The DOLCIANI MATHEMATICAL EXPOSITIONS series of the Mathe­ matical Association of America was established through a generous gift to the Association from Mary P. Dolciani, Professor of Mathematics at Hunter College ofthe City University ofNew York. In making the gift, Professor Dolciani, herself an exceptionally talented and successful expositor ofmathematics, had the purpose of furthering the ideal of excellence in mathematical exposition. The Association, for its part, was delighted to accept the gracious gesture initiating the revolving fund for this series from one who has served the Association with distinction, both as a member of the Committee on Publications and as a member of the Board of Governors. It was with genuine pleasure that the Board chose to name the series in her honor. The books in the series are selected for their lucid expository style and stim­ ulating mathematical content. Typically, they contain an ample supply ofexercises, many with accompanying solutions. They are intended to be sufficiently elementary for the undergraduate and even the mathematically inclined high-school student to understand and enjoy, but also to be interesting and sometimes challenging to the more advanced mathematician. 1. Mathematical Gems, Ross Honsberger 2. Mathematical Gems IL Ross Honsberger 3. Mathematical Morsels, Ross Honsberger 4. Mathematical Plums, Ross Honsberger (ed.) 5. Great Moments in Mathematics (Before 1650), Howard Eves 6. Maxima and Minima without Calculus, Ivan Niven 7. Great Moments in Mathematics (After 1650), Howard Eves 8. Map Coloring, Polyhedra, and the Four-Color Problem, David Barnette 9. Mathematical Gems IlL Ross Honsberger 10. More Mathematical Morsels, Ross Honsberger 11. Old and New Unsolved Problems in Plane Geometry and Number Theory, Victor Klee and Stan Wagon 12. Problems for Mathematicians, Young and Old, Paul R. Halmos 13. Excursions in Calculus: An Interplay of the Continuous and the Discrete, Robert M. Young 14. The Wohascum County problem Book, George T. Gilbert, Mark Krusemeyer, and Loren C. Larson 15. Lion Hunting and Other Mathematical Pursuits: A Collection ofMathematics, Verse, and Stories by Ralph P. Boas, Jr., edited by Gerald L. Alexanderson and Dale H. Mugler 16. Linear Algebra Problem Book, Paul R. Halmos 17. From Erdos to Kiev: Problems ofOlympiad Caliber, Ross Honsberger To Nancy CONTENTS Preface xi Morsel 1. A Surprising Property of the Integer 11 1 Morsel 2. An Unexpected Equality 4 Morsel 3. An Arithmetic Game 6 Morsel 4. Sangaku 8 Morsel 5. Pagan Island 13 Morsel 6. Persistent Numbers 15 Gleanings from Murray Klamkin's Olympiad Corners-1979 19 Morsel 7. An Application ofvan der Waerden's Theorem 28 Morsel 8. Patruno's Proof of a Theorem of Archimedes 31 Morsel 9. An Intractable Sum 33 Morsel 10. A Cyclic Quadrilateral 36 Gleanings from Murray Klamkin's Olympiad Corners-1980 38 Morsel 11. An Unlikely Perfect Square 52 Morsel 12. An Unlikely Symmetry 54 Morsel 13. Two Famous Diophantine Equations 57 Morsel 14. Brianchon and Ceva 61 Morsel 15. Playing on a Polynomial 64 Morsel 16. Overlapping Quadrants 67 Morsel 17. The Circle and the Annulus 70 Gleanings from Murray Klamkin's Olympiad Corners-1981 73 Morsel 18. On a Balanced Incomplete Block Design 103 Morsel 19. The Red and White Balls 106 Morsel 20. A Prime Number Generator 108 Morsel 21. Neuberg's Theorem 110 Morsel 22. A Geometric Calculation 112 Morsel 23. On Cubic Curves 114 Morsel 24. An Olympiad Practice Problem 119 Gleanings from Murray Klamkin's Olympiad Corners-1982 122 Morsel 25. A Quadruple of Consecutive Integers 131 Morsel 26. A Box-Packing Problem 133 Morsel 27. An Awkward Integral 136 Morsel 28. A Matter of Perspective 138 Morsel 29. Sequences of Nested Radicals 140 Morsel 30. Equations of Factorials 145 Gleanings from Murray Klamkin's Olympiad Corners-1983 146 Morsel 31. An Oft-Neglected Form ,.171 Morsel 32. A Theorem of Leon Anne 174 Morsel 33. Special Pairs of Positive Integers 176 Morsel 34. An Intriguing Sequence 178 Morsel 35. That Number Again 180 Morsel 36. A Rational Function 183 Morsel 37. An Unexpected Bijection 185 Morsel 38. An Unruly Sum 187 Gleanings from Murray Klamkin's Olympiad Corners-1984 190 Morsel 39. An Interesting Inequality 205 Morsel 40. A Series of Reciprocals 207 Morsel 41. On the Least Common Multiple 209 Morsel 42. A Family of Equations 212 Gleanings from Murray Klamkin's Olympiad Corners-1985 215 Morsel 43. Diophantine Reciprocals 232 Morsel 44. Another Series of Reciprocals ~ 234 Morsel 45. An Illegible Multiple-Choice Problem 235 Morsel 46. On the Partition Function p(n) 237 Gleanings from Murray Klamkin's Olympiad Corners-1986 240 Morsel 47. Proof by Interpretation 246 Morsel 48. On a(n) and T(n) 250 Morsel 49. A Surprising Result About Tiling a Rectangle 252 Morsel 50. An Amazing Locus 261 MorselSI. Moessner's Theorem 268 Morsel 52. Counting Triangles 278 Morsel 53. A Geometric Mimimum 283 Morsel 54. The Probabilistic Method 286 Morsel 55. An Application of Generating Functions 292 Morsel 56. On Rat-Free Sets 299 Morsel 57. A Further Note on Old Morsel 23 312 References 319 PREFACE Entitling this book More Mathematical Morsels has the advantage ofimme­ diately informing anyone who is acquainted with my Mathematical Morsels just what kind ofbook to expect. Ifit hadn't been for the earlier volume, the present collection might well have been called Gleanings From Crux Math­ ematicorum, referring to a wonderful undergraduate problems journal that was founded in Ottawa, Canada, in 1975 by Leo Sauve and Fred Maskell. Except for a few miscellaneous cases, I encountered all the problems dis­ cussed here in Crux Mathematicorum. Of particular interest and value are the. many slates of problems given by Murray Klamkin in his regular col­ umn "Olympiad Comer" during the years 1979-1986. There is much for every level of mathematics scholar to enjoy in Crux Mathematicorum­ the examples touched on here represent only a small fractidn of the good things to be found there. In this collection we shall be concerned only with elementary prob­ lems, and, ofthose, only ones that aren't too long or complicated. Needless to say, I hope that each ofthem contains something exciting-a surprising result, an intriguing approach, a stroke of ingenuity. They are not pre­ sented in any attempt to be instructive, but solely in the hope of giving enjoyment-an elegant or ingenious argument can be a beautiful thing.. In vindication of the original contributors to Crux Mathematicorum, it should be noted that I have written up these brief essays to suit myself, and have generally raided their work as I pleased; consequently, I have only myself to blame for all the faults in the result. xi xii MORE MATHEMATICAL MORSELS I remember reading somewhere years ago that J. L. Synge declared that "The mind is at its best when atplay." I have always felt there might be something to this idea and so I hope that you will approach these vignettes as mathematical entertainment. REFERENCES All references, except for a very few that are specifically noted, are to the journal Crux Mathematicorum, published by the Canadian Mathematical Society. Morsel 1. 1979, 102, Olympiad Comer 4, Sample Problem from the Mathematics Student. Morsel 2. 1983, 56, Problem 712, proposed by Donald Aitken, Northern Alberta Institute of Technology, Edmonton, Alberta; solved by Bernard Baudiffier, College de Sherbrooke, Sherbrooke, Quebec. Morsel 3. 1980, 17, Problem 418, proposed by James Gary Propp, student, Harvard College; solved by Leroy F. Meyers, The Ohio State University. Morsel 4. 1986, 58, Problem 995, proposed by Hidetosi Fukagawa, Yokosuka High School, Tokai-City, Aichi, Japan; solved by Sam Baethge, San Antonio, Texas. Morsel 5. 1982,325, Problem 705, proposed by Andy Liu, University of Alberta; solved by Leroy F. Meyers, The Ohio State University. Morsel 6. 1981, 185, Problem 555, proposed by Michael W. Ecker, Pennsylvania State University, Worthington Scranton Campus; solved by John T. Barsby, St. John's-Ravenscourt School, Winnipeg, Manitoba. Morsel 7. 1984, 157, Problem 817, solved by Stanley Rabinowitz, Digital Equip­ ment Corporation, Merrimack, New Hampshire; solved by Murray Klamkin and A. Meir, University of Alberta. Morsel 8. 1980, 188, Problem 466, proposed by Roger Fischler, Carleton University, Ottawa; solved by Gregg Patruno, student, Stuyvesant High School, New York. Morsel 9. 1976, 80-81, Problem 108, proposed by Viktors Linis, University of Ottawa; solved by F. G. B. Maskell, Algonquin College. Morsel 10. 1983, 221, Problem 759, proposed by Jack Garfunkel, Flushing, New York; solved by Leon Bankoff, Los Angeles, California; comment by Leroy F. Meyers, The Ohio State University. Morsel 11. 1976, 95, Problem 111, proposed by H. G. Dworschak, Algonquin College; solved by Leo Sauve , Algonquin College. Morsel 12. 1977,143, Problem 206, proposed by Dan Pedoe, University of Min­ nesota; solved by Dan Sokolowsky, Yellow Springs, Ohio. Morsel 13. Part (b): 1977, 73, Problem 188, proposed by Daniel Rokhsar, Susan Wagner High School, Staten Island, New York; solved by W.
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