Problems from Murray Klamkin

Andy Lui and Bruce Shawyer,Editors

The Canadian Collection

AMS / MAA PROBLEM BOOKS VOL 21

Problems from Murray KlamkinThe Canadian Collection

© 2009 byThe Mathematical Association ofAmerica (Incorporated)

Library ofCongress Catalog Card Number 2008939094

ISBN: 978-0-88385-828-8

Printed in the United States ofAmerica

Current Printing (last digit):10 9 8 7 6 5 4 3 2 1

Problems from Murray KlamkinThe Canadian Collection

Edited by

AndyLiu

and

Bruce Shawyer

Published byMathematical Association of America

in collaboration with the.Canadian Mathematical Society

10.1090/prb/021

Council on PublicationsPaul Zorn, Chair

Problem Books Editorial Board

Richard A. Gillman Editor

Roger NelsenMark Saul

Tatiana Shubin

MAA PROBLEM BOOKS SERIES

Problem Books is a series of the Mathematical Association of America consisting of col­lections of problems and solutions from annual mathematical competitions; compilationsof problems (including unsolved problems) specific to particular branches of mathematics;books on the art and practice of problem solving, etc.

Aha! Solutions, Martin EricksonThe Contest Problem Book VII: American Mathematics Competitions, 1995-2000 Contests, com­

piled and augmented by Harold B. ReiterThe Contest Problem Book VIII: American Mathematics Competitions (AMC 10), 2000-2007, com­

piled and edited by J. Douglas Faires & David Wells

The Contest Problem Book IX: American Mathematics Competitions (AMC 12), 2000-2007, com-piled and edited by David Wells & J. Douglas Faires

A Friendly Mathematics Competition: 35 Years ofTeamwork in Indiana, edited by Rick Gillman

The Inquisitive Problem Solver, Paul Vaderlind, Richard K. Guy, and Loren C. Larson

International Mathematical Olympiads 1986-1999, Marcin E. Kuczma

Mathematical Olympiads 1998-1999: Problems and Solutions From Around the World, edited byTitu Andreescu and Zuming Feng

Mathematical Olympiads 1999-2000: Problems and Solutions From Around the World, edited byTitu Andreescu and Zuming Feng

Mathematical Olympiads 2000-2001: Problems and Solutions From Around the World, edited byTitu Andreescu, Zuming Feng, and George Lee, Jr.

Problems from Murray Klamkin: The Canadian Collection, edited by Andy Liu and Bruce Shawyer

The William Lowell Putnam Mathematical Competition Problems and Solutions: 1938-1964, A. M.Gleason, R. E. Greenwood, L. M. Kelly

The William Lowell Putnam Mathematical Competition Problems and Solutions: 1965-1984, GeraldL. Alexanderson, Leonard F. Klosinski, and Loren C. Larson

The William Lowell Putnam Mathematical Competition 1985-2000: Problems, Solutions, and Com-mentary, Kiran S. Kedlaya, Bjorn Poonen, Ravi Vakil

USA and International Mathematical Olympiads 2000, edited by Titu Andreescu and Zuming Feng

USA and International Mathematical Olympiads 2001, edited by Titu Andreescu and Zuming Feng

USA and International Mathematical Olympiads 2002, edited by Titu Andreescu and Zuming Feng

USA and International Mathematical Olympiads 2003, edited by Titu Andreescu and Zuming Feng

USA and International Mathematical Olympiads 2004, edited by Titu Andreescu,Zuming Feng, and Po-Shen Loh

MAA Service Center~ O. Box 91112

Washington, DC 20090-11121-800-331-1622 fax: 1-301-206-9789

Murray Seymour K1amkin (1921-2004)

Memorial Celebrationof the Life of Murray Klamkin*

Let me first make it clear that this is not a eulogy. By my definition, a eulogy is an attemptto make the life of the departed sound much better than it was. In the present case, itis not only unnecessary, it is actually impossible. Murray Seymour Klamkin had a mostproductive and fulfilling life, divided between industry and academia.

Of the early part of his life, I knew little except that he was born in 1921 in Brooklyn,New York, where his father owned a bakery. This apparently induced in him his life-longfondness for bread. I read in his curriculum vita that his undergraduate degree in ChemicalEngineering was obtained in 1942 from Cooper Union's School of Engineering. Duringthe war, he was attached to a chemical warfare unit stationed in Maryland, as his youngersister Mrs. Judith Horn informed me.

In 1947, Murray obtained a Master of Science degree from the Polytechnic Institute ofNew York, and taught there until 1957 when he joined AVCO's Research and AdvancedDevelopment Division. In 1962, Murray returned briefly to academia as a professor atSUNY, Buffalo, and then became a visiting professor at the University of Minnesota. In1965, he felt again the lure of industry and joined Ford Motor Company as the PrincipalResearch Scientist, staying there until 1976.

During all this time, Murray had been extremely active in the field of mathematics prob­lem solving. His main contribution was serving as the editor of the problem section ofSIAM Review. He had a close working relation with the Mathematical Association ofAmerica, partly arising from his involvement with the William Lowell Putnam Mathemat­ics Competition.

In 1972, the MAA started the USA Mathematical Olympiad, paving the way for thecountry's entry into the International Mathematical Olympiad in 1974, hosted by what wasstill East Germany. Murray was unable to obtain from Ford release time to coach the team.Disappointed, he began to look elsewhere for an alternative career. This was what broughthim to Canada, at first as a Professor of Applied Mathematics at the University of Waterloo.However, it was not until the offer came from the University ofAlberta that he made up his

*This memorial tribute first appeared in the special issue of Crux Mathematicorum dedicated to Murray Klamkinin September, 2005.

vii

viii Problems from Murray Klamkin

mind to leave Ford. I do not know if Murray had been to Banff before, but he must havevisited this tourist spot during the negotiation period, fell in love with the place and closedthe deal.

As Chair, Murray brought with him a management style from the private sector. Appar­ently not everyone was happy with that, but he did light some fires under several pairs ofpants, and rekindled the research programs of the wearers.

Murray had always been interested in Euclidean Geometry. He often told me about hishigh school years when he and a friend would challenge each other to perform variousEuclidean constructions. Although the Chair had no teaching duties at the time, Murraytook on a geometry class himself.

At the same time, Murray began editing the Olympiad Comer in Crux Mathematicorum,a magazine then published privately by Professor Leo Sauve of Ottawa. It is now an officialjournal of the Canadian Mathematical Society. Murray also introduced the Freshmen andUndergraduate Mathematics Competitions in the Department.

Geometry, mathematics competitions and Crux Mathematicorum were what brought meto Murray's attention. At the time, I was a post-doctoral fellow seeking employment, hav­ing just graduated from his Department. Thus I was ready to do anything, and it happenedth~t my interests coincided with those of Murray. I was holding office hours for his geom­etry class, helping to run the Department's competitions and assisting him in his editorialduties.

I remember being called into his office one day. He had just received a problem proposalfor Crux Mathematicorum. "Here is a nice problem," he said, "but the proposer's solutionis crappy. Come up with a nice solution, and I need it by Friday afternoon!"

As much as I liked problem-solving, I was not sure that I could produce results by anindustrial schedule. Nevertheless, I found that I did respond to challenges, and although Iwas not able to satisfy him every time, I managed to do much better than if I was left onmy own, especially after I had got over the initial culture shock.

The late seventies were hard times for academics, with few openings in post-secondaryinstitutions. I was short-listed for every position offered by the Department, but alwayscame just short. Eventually, I went elsewhere for a year as sabbatical replacement. Murraycame over to interview me for a new position, pushed my appointment through the HiringCommittee and brought me back in 1980.

Murray had been the Deputy Leader for the USA National Team in the IMO since 1975.In 1981, USA became the host of the event, held outside Europe for the first time. SamGreitzer, the usual Leader, became the chief organizer. Murray took over as the Leader,and secured my appointment as his Deputy Leader.

I stayed in that position for four year, and in 1982, made my first trip to Europe be­cause the IMO was in Budapest. This was followed. by IMO 1983 in Paris, and IMO1984 in Prague. I was overawed by the international assembly, but found that they in tumwere overawed by Murray's presence. He was arguably the most well-known mathematicsproblem-solver in the whole world.

We both retired from the IMO after 1984, even though I would later return to it. His termas Chair also expired in 1981. Thus our relationship became collegial and persotlal. He andhis wife Irene had no children, but they were very fond of company. I found myself a guestat their place at regular interval, and they visited my humble abode a few times.

Memorial Celebration ix

It was during this period that I saw a different side of Murray. Before, I found him verybusinesslike, his immense talent shining through his incisive insight and clinical efficiency.Now I found him a warm person with many diverse interest, including classical music,ballroom dancing, adventure novels, kung-fu movies and sports, in particular basketball.

Although Murray had been highly successful in everything he attempted, he will prob­ably be remembered the most for his involvement in mathematics problem-solving andcompetitions. He had authored or edited four problem books, and had left his mark in ev­ery major journal which had a problem section. He had received an Honorary Doctoratefrom the University of Waterloo and was a Fellow of the Royal Society of Belgium. Hehad won numerous prizes, and had some named after him.

Murray had enjoyed remarkably good health during his long life. It began to deterioratein September 2000 when he underwent a by-pass operation. After his release from thehospital, he continued to exert himself, walking up to his office on the sixth floor, andskating in the West Edmonton Mall.

His heart valve gave in November, fortunately while he was already in the hospital forphysiotherapy. He was in a coma for some time. One day, when I visited him, he wasbleeding profusely from his aorta. The doctor indicated to me that he did not expect Murrayto la~t through the day.

Somehow, the inner strength of Murray came through, and on my next visit, he was fullyconscious. He told me to make arrangements for his eightieth birthday party, stating simplythat he would be out of the hospital by that time. It was a good thing that I took his wordsseriously, for he was out of the hospital by that time, ready to celebrate.

One of the last mathematical commitments he made was to edit the problem section inthe MAA's new journal Math Horizons. During this difficult time, he asked me to servewith him as joint-editors. Later, he passed the column onto me, but his finger-prints werestill allover the pages.

Now I have to try to fill his shoes without the benefit of his wisdom. His passing marksthe end of an era in the world of mathematics competition and problem-solving. He willbe deeply missed.

Andy LiuDepartment of Mathematical and Statistical SciencesUniversity of AlbertaEdmonton, Alberta

Acknowledgements

We would like to thank our co-publishers Jonathan Borwein of the Canadian Mathe­matical Society and Don Albers of the Mathematical Association of America, as well asour copy editor Beverly Ruedi of the MAA. We are grateful to David Langstroth andCraig Platt of the CMS for administrative and technical assistance. Solutions from pre­Jb.TEX editions of Crux Mathematicorum were coded by students at Memorial University ofNewfoundland, funded by the eMS. Special thanks are due to Karelyn Davis, Alyson Ford,Don Hender, Shawna Gammon, Paul Marshall, Shannon Sullivan and Rebecca White.

Since all the material from this book comes from Crux Mathematicorum, we must alsopay tribute to its founding editors Leo Sauve and Fred Maskell, past editors Bill Sands,Robert Woodrow, Bruce Shawyer, and Jim Totten, and current editor Vaclav Linekand the many members of the Board of Crux Mathematicorum over the years, who havemoderated problems and solutions from our contributors world-wide. We are most gratefulto Graham Wright for close to thirty years of service as the managing editor, on top ofbeing the Executive Director of the Canadian Mathematical Society.

InformationFor a subscription to Crux Mathematicorum with Mathematical Mayhem, write to

The Canadian Mathematical Society,577 King Edward Avenue,Ottawa, Ontario, KIN6N5, Canada,

or check their website at http://cms.math.ca.

xi

Introduction

When one thinks of mathematical problems posed over the past 30 years in Canada (andelsewhere), one name automatically leaps to the fore - Murray Klamkin.

Murray's first foray into the Canadian scene came in 1977, three years after he arrived inCanada. He posed a problem in EUREKA (now Crux Mathematicorum with MathematicalMayhem). He continued to proposed problems right up to 2001, when he was 80 yearsold. In that year, Crux Mathematicorum with Mathematical Mayhem dedicated a completeissue to celebrate his 80th birthday. Although he was seriously ill, he continued to poseproblems, solve problems and make shrewd comments on other people's proposals andsolutions.

Murray made important contributions to the IMO in Canada in 1995. In particular, hewas the judge for the prize awarded to a student for an outstanding and innovative solutionto a problem. As Editor-in-Chief of Crux Mathematicorum with Mathematica'z Mayhemfrom 1996 to 2002, I had much correspondence with Murray, and really learned the beautyof his mind.

Throughout his entire career in the United States and in Canada, Murray was in theforefront of the problemist community. He worked with some of the best students in bothcountries and left a very indelible mark. This book is dedicated to Murray and to the pro­found influence that he has had on many, many people allover the world.

Murray Klamkin was famous for his Quickies, problems that had quick and neat solu­tions. In this book, we present all the Quickies published in Crux Mathematicorum, withsome slight editing by Bruce Shawyer.

We also present all the problems Murray Klamkin had proposed for Crux Mathemati­corum. These problems exhibit the special talents of Murray Klamkin. They cover a verywide range of topics, and show a great deal of insight into what is possible in these areas.They demonstrate that Murray Klamkin was a problem setter par excellence. We are verygrateful to have known him and to have been inspired by him.

The problems have been arranged into sets according to topic by Andy Liu. The solutionsare as published in Crux Mathematicorum, with some slight editing by Bruce tihawyer.When a problem number is followed by a star *,this means that the problem was proposedwithout a solution.

xiii

xiv Problems from Murray Klamkin

Finally, we present some problems proposed by others in tribute to Murray Klamkin.Unfortunately, he died in 2004. His contributions are sadly missed.

Bruce Shawyer,

Department of Mathematics,Memorial University of Newfoundland,St. John's, Newfoundland and Labrador.

Contents

Memorial Celebration of the Life of Murray K1amkin

Acknowledgements

Introduction

Problems1 Quickies

2 Klamkin's Problems in Crux Mathematicorum1 Combinatorics and Number Theory .

2 Functions and Polynomials.....

3 Expressions and Identities .

4 Numerical Approximations

5 Algebraic Inequalities

6 Trigonometric Inequalities .

7 Geometric Inequalities .

8 The Triangle

9 Cevian Lines .

10 Central Symmetry

11 Conic Sections .

12 Solid Geometry

13 Higher Dimensions14 Vectors and Matrices .

15 Calculus

3 The KIamkin Tribute Problems-2005

II Solutions

4 Quickies

vii

xi

xiii

1

3

1111

131516

1720

2324

26

27

28

29303234

37

41

43

xv

Combinatorics and Number Theory .

xvi

5 K1amkin's Problems in Crux Mathematicorum12 Functions and Polynomials3 Expressions and Identities .4 Numerical Approximations5 Algebraic Inequalities . . .6 Trigonometric Inequalities .7 Geometric Inequalities .8 The Triangle . . .9 Cevian Lines . . .10 Central Symmetry11 Conic Sections . .12 Solid Geometry .13 Higher Dimensions14 Vectors and Matrices. .15 Calculus .

6 The K1amkin Tribute Problems-2005

Index of Proposers and Solvers

Index of Crux Problems

About tbe Editors

Problems from Murray Klamkin

616170818992

115151159170178184186199211220

225

241245

249

Index of Proposers and Solvers

Aassila, Mohammed, Strasbourg, France, 39Abbott, Harvey, 12Ahlburg, Hayo, Benidorm, Spain, 66, 154Areias, Wilson da Costa Rio de Janeiro, Brazil,

84, 166

Baethge, Sam, Science Academy, Austin, TX, 83Bang, Seung-Jin, Seoul, Republic ofKorea, 84,

215Bankoff, Leon, Los Angeles, CA, 167, 186Baron, Gerd, Technische Universitiit Wien, Aus-

tria, 87, 172Bataille, Michel, Rouen, France, 39, 112,235Bencze, Mihaly, Brasov, Romania, 38, 39Benito, Manuel, Logroiio, Spain, 225,228, 230,

238Benito, Maria Mercedes Sanchez, I.B. Luis

Bunuel, Alcorcon, Madrid, Spain, 138Bilchev, Svetoslav J., Technical University,

Russe, Bulgaria, 135Blundon, W.J., Memorial University of New­

foundland, St. John's, 159,162,164Bluskov, Iliya, Technical University, Gabrovo,

Bulgaria, 85Boente, P., 201Bornsztein, Pierre, Pontoise, France, 113Bos, Len University of Calgary, Calgary, Al­

berta, 73, 77Bosley, Carl, Washburn Rural High School,

Topeka, KS, 69Bottema, 0., 22Bradley, ChristopherJ., Clifton College, Bristol,

UK, 203,217Brown, Scott, Auburn University, Montgomery,

AL,237

Cirtoaje, Vasile, University of Ploiesti, Roma­nia, 38

Campbell, Paul J .,for the Beloit College SolversBeloit, WL 220

Cautis, Sabin, Earl Haig Secondary School,North York, Ontario, 108

Chambers, G.A., 32Chan, Adrian, Upper Canada College, Toronto,

Ontario, 70Chang, Andre, Woburn Collegiate, Scarbor­

ough, Ontario, 93Choudhury, Himadri, Hunter High School, New

York, NY, 150Ciaurri, Oscar, Logroiio, Spain, 225, 228, 230,

238Cooper, Curtis, Central Missouri State Univer­

sity, 225, 227, 235

Diaz-Barrero, Jose Luis, Universitat Politecnicade Catalunya, Barcelona, Spain, 38

Das, Jayabrata , Calcutta, India, 79Denham, Graham , University of Alberta, Ed­

monton, 202Diminnie, Charles R., Angelo State University,

San Angelo, TX, 114Dou, Jordi, Barcelona, Spain, 152, 183, 185

Engelhaupt, Hans, Franz-Ludwig-Gymnasium,Bamberg, Germany, 185

Englander, Mathew, Kitchener, Ontario, 65Eves, Howard, University of Maine, Orono,

184, 188

Fernandez Moral, Emilio, I.B. Luis Bunuel, Al­corcon, Madrid, Spain, 138

241

242

Fernandez, Emilio, Logrofio, Spain, 225, 228,230,238

Ferrari, LB. Emilio, Valladolid, Spain, 84Festraets-Hamoir, C., Brussels, Belgium, 63,

137Fisher, Benji, New York, NY, 170Fritsch, Rudolf, Munich, Germany, 203Fukagawa, Hidetosi, Yokosuka High School,

Tokai-City, Aichi, Japan, 153

Garfunkel, Jack, Flushing, NY, 17, 26, 92, 94,117

Gee, Toby, the John of Gaunt School, Trow­bridge, England, 69

Gislason, Gary, University of Alaska, Fair­banks, Alaska, 34,221

Godin, Shawn, St. Joseph Scollard Hall, NorthBay, Ontario, 86

Groenman, J.T., Arnhem, the Netherlands, 178

Harterich, Jorg, Winnenden, Federal RepublicofGermany, 76

Hanjs, Zeijko, University of Zagreb, Zagreb,Croatia, 113

Henderson, G.P. , Campbellcroft, Ontario, 83,107,155,190,211

Hersonsky, Sa'ar, Technion, Haifa, Israel, 95Hess, Richard I., Rancho Palos Verdes, CA, 66,

69,92,125,128,212Higham, Jeff, University ofToronto, 101, 147Howard, Joe, Portales, NM, 113Hsia, Cyrus, University of Toronto, Toronto,

Ontario, 80Huang, Jun-hua, The 4th Middle School of

Nanxian, Hunan, China, 148Huang, Jun-hua, the Middle School attached to

Hunan Normal University, Changsha,China, 106

Israel, Robert B., University of BritishColumbia, Vancouver, BC, 76, 212,229

Janous, Walther, Ursulinengymnasium, Inns­bruck, Austria, 37,69,95,98,101, 102,114,115,117,118,122,135,196,221,.222,232,233

Konecny, Vaclav, Ferris State University, BigRapids, MI, 69, 168

Kuczma, Marcin E., Warszawa, Poland, 64,102,104,128,174,182,183,195,202

LaMacchia, Sam, Yellow Springs, OH, 163

Index of Proposers and Solvers

Lau, Kee-Wai, Hong Kong, 69,85,90, 104,149,167

Lausch, Hans, Monash University, Melbourne,Australia, 214

Lewis, Kathleen E., SUNY, Oswego, NY, 69Ling, Joseph, University of Calgary, Calgary,

Alberta, 202Liu, Andy, University ofAlberta, Edmonton, Al­

berta, 11,17,25,28,30,40,61,100,189

Liu, Henry, University of Memphis, Memphis,TN, 177

Loeffler, David, Trinity College, Cambridge,UK, 169,210

Maragoudakis, Pavlos, University of Athens,Greece, 102

Maskell, F.G.B., College Algonquin, OttawaOntario, 72

Matizen, V.E., 101McCartney, Phil, Northern Kentucky University,

Highland Heights, KY, 39Meyers, Leroy F., Ohio State University,

Columbus, 67, 70, 154, 184, 213Munoz, Manuel Benito, I.B. Luis Bunuel, AI­

corc6n, Madrid, Spain, 138Murty, Vedula N., Pennsylvania State Univer­

sity, Capitol Campus, PA, 93Marquez, Juan Bosco Romero, Universidad de

Valladolid, Valladolid, Spain, 39, 111

Newman, DJ., Temple University, Philadel­phia, PA, 35

Paragiou, Theoklitos, Limassol, Cyprus,Greece, 176

Penning, P., Delft, the Netherlands, 68, 75, 180,216,223

Petrov, Fedor, Saint Petersburg, Russian Feder­ation, 203

Pompe, Waldemar, University of Warsaw,Poland, 157

Rabinowitz, Stanley, Westford, MA, 65Rennie, Basil C., James Cook University of

North Queensland, Townsville, Aus­tralia, 92

Romero, Carlos, I.B. Manuel Blancafort, LaGarriga, Catalonia, Spain, 213

Rosado, Francisco Benot, I.B. Emilio Ferrari,Valladolid, Spain, 84, 216

Sagasta, LB., Logrofio, I.B. Luis Bunuel, AI­corc6n, Madrid, Spain, 138

Index of Proposers and Solvers

Salvatore, Gali, Perkins, Quebec, 81Sands, Bill, University ofCalgary, Calgary, Al-

berta, 40, 73Sastry, K.R.S., Bangalore, India, 26Sauve, Leo, 20, 23, 34Schlosberg, Joel, Bayside, NY, 197, 236, 239Schwarz, Sibylle, Friedrich-Schiller-

Universitiit, lena, Germany, 156Seiffert, Heinz-Jiirgen, Berlin, Germany, 68,69,

107,111Seimiya, Toshio, Kawasaki, lapan, 26, 27Shafer, Robert E., Berkeley, CA, 66,214Shapiro, Daniel B., Ohio State University,

Columbus, 213Smith, Malcolm A., Georgia Southern College,

Statesboro, 171Sokolowsky, Dan, Antioch College, Yellow

Springs, OH, 159, 162, 163Specht, Eckard, Otto-von-Guericke University,

Magdeburg, Germany, 113Springer, Colin, University ofWaterloo, Water­

loo, Ontario, 91, 222Subbarao, M.V., University ofAlberta, Edmon­

ton, Alberta, 13, 31 69

Tabov, Jordan B., Sofia, Bulgaria, 89

243

Thuan, Pham Van, Hanoi University ofScience,Hanoi, Vietnam, 38

Tsintsifas, George, ThessaIoniki, Greece, 40,151,165,199,201

van de Craats, Jan, Leiden University, TheNetherlands, 170

Varberg, Dale, Hamline University, St. Paul,Minnesota, 30, 196

Vlachakis, John, Athens, Greece, 218

Waldmann, Johannes, Friedrich-Schiller­Universitiit, lena, Germany, 156

Wang, Edward T.H., Wilfrid Laurier University,Waterloo, Ontario, 220

Wayne, Alan, Pasco-Hernando CommunityCollege, New Port Richey, FL, 164

Westbrook, Rex, University of Calgary, Cal­gary, Alberta, 78

Wildhagen, C., Breda, the Netherlands, 222Wildhagen, Chris, Rotterdam, the Netherlands,

108

Zhao, Yufei, Don Mills Collegiate Institute, 228Zhou, Li, Polk Community College, Winter

Haven, FL, 38, 113,238,240 115,226,233

Index of Crux Problems

210,24,159 1062,27,178224,30,199 1076,25,165254,13,70 1086,31,199273,34,220

1110:14, 73287,15,81299,14,72 1131,23,152

1147,34,220323,17,92 1165:23, 152330:29, 186 1178,34,221347,17,93 1200,32,212375,29,187 1207,32,213398,32,211 1213:16,90

429,11,61 1242,32,213

478,29,188 1261:29, 190

485,26,170 1271,20,1181283,14,76

506,23,151 1296,23,153520,28,184

1304,15,83548,26,1711314,33,214

805,17,93 1322,35,222830,15,82 1332,21,122

1348:27, 180908,20,115 1352,21,124958,20,117 1362,15,83969,11,62 1371:16,91

1003:16,891385,25,166

1027,11,631394,17,94

1060,20,117 1405,28,185

245

246 Index of Crux Problems

1414,21,135 1945,24,1561423:14,76 1962,22,1501434,12,64 1975:28, 1851445, 17, 100 1985,24,1571456,12,65 1996,16,851465,31,2021473:24, 154 2005,33,217

1482,33,214 2014, 14, 79

1494,35,223 2024,31,2032034,13,68

1503,21,137 2044, 19, 1071513,27,183 2054, 13, 701522,15,84 2064, 19, 1081532,25,167 2084,22,1501542:21, 138 2095,19,1111553,29,1951561,12,66 2613,26,176

1574,24,154 2614,27,177

1581:30, 196 2615,19,111

1594,16,85 2616:28, 1862617,30,197

1605,25,167 2618,25,1681613,22,147 2651:31,2031621:26, 1721631:26, 174 2733:32,2101642, 17, 101 2734,19,112

1652, 18, 1022839,20,114

1662, 18, 1021674, 18, 104 2848,26,169

1693,33,216 3024,34,219

1703, 18, 104 K-Ol, 37,2251712,22,148 K-02, 37, 2271721,33,216 K-03, 38, 2281734, 18, 105 K-04, 38, 2291742, 18, 106 K-05, 38, 2301752:12,67 K-06, 38, 2301764,24,155 K-07, 38, 2321774:19, 107 K-08, 39,2331784,30,196 K-09, 39, 2351793,31,202 K-I0, 39, 236

1801,22,149 K-l1, 39, 237

1863,13,68 K-12, 39, 238

1872:25, 168 K-13, 40,238K-14, 40, 239

1935,28,185 K-15, 40, 240

Index of Crux Problems

Q-l, 3,43Q-2, 3,43Q-3, 3,43Q-4, 3,44Q-5, 3,45Q-6, 4, 46Q-7, 4, 46Q-8, 4, 46Q-9, 4, 46Q-IO, 4, 47Q-ll,4, 47Q-12, 4,48Q-13, 5, 48Q-14, 5, 48Q-15, 5, 49Q-16, 5, 49Q-17, 5, 49Q-18, 5,50Q-19, 5, 50Q-20, 5,50Q-21, 5, 51Q-22, 6,51Q-23, 6,51Q-24, 6,52Q-25, 6,52Q-26, 6,53Q-27, 6,53

Q-28, 6,54Q-29, 6,54Q-30, 7, 54Q-31, 7, 55Q-32, 7,55Q-33, 7, 55Q-34, 7, 55Q-35, 7, 55Q-36, 7, 56Q-37, 7,56Q-38, 7,56Q-39, 8,57Q-40, 8,57Q-41, 8,57Q-42, 8,57Q-43, 8,57Q-44, 8,57Q-45, 8,58Q-46, 8,58Q-47, 8,58Q-48, 9,58Q-49, 9,58Q-50, 9, 59Q-51, 9, 59Q-52, 9, 59Q-53, 9,59Q-54, 9,60

247

About the Editors

Andy Lin received a as degree with First Class Honors in Mathematics from McGill Uni­versity in 1970. He earned his MS in Number Theory (1972) and a Doctor ofPhilosophy inCombinatorics (1976) from the University of Alberta, Edmonton. Dr. Liu won a MinnesotaMining & Manufacturing Teaching Fellowship in 1998. He has won numerous teachingawards including: Canadian University Professor of the Year in 1998 (awarded by theCanadian Council for the Advancement ofEducation and the Council for the Advancementand Support of Education); the Canadian Mathematical Society's Adrien Pouliot Educa­tion Award in 2003; and the Mathematical Association of America's Deborah and FranklinTepper Haimo Teaching Award in 2004. Dr. Liu is very well known in problem-solving cir­cles. He won the David Hilbert International Award in 1996 from the World Federation ofNational Mathematics Competitions. He was the Deputy Leader (under Murray Klamkin)of the USA Mathematical Olympiad Team from 1981-1984; he later was Leader of theCanadian IMO team. Dr. Liu has been the Editor of the problem section of the MAA'sMath Horizons for seven years.

Bruce Shawyer earned his Ph.D. in Mathematics from the University of St. Andrews inthe UK in 1963. He taught at the University of Nottingham from 1962-66 and at the Uni­versity of Western Ontario from 1966-85. In 1985 he joined Memorial University as fullprofessor, serving as "head of the Department of Mathematics and Statistics from 1985­91. Dr. Shawyer's professional interests in mathematics research include approximationof the sums of slowly convergent series, the summability of series and integrals, and Eu­clidean geometry. In the area of mathematics education he is interested in mathematicsenrichment, why certain mathematical ideas are not well-learned and the propagation ofmathematics. He is founder of the mathematical challenge for junior high school studentsand former head coach of the Canadian team for the International Mathematical Olympiad.Dr. Shawyer has authored 118 publications and is currently Editor-~n-Chiefof the ATOM(A Taste Of Mathematics) book series. He has taught mathematics courses at all levels.

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Problems from Murray KlamkinAndy Lui and Bruce Shawyer, Editors

Murray Klamkin was a dedicated problem solver and problem proposer, who left indelible marks on the problemist community. After working in industry and academe in the United States, he spent the last 30 of his 84 years in Canada. He was famous for his Quickies, problems that have quick and neat solutions. In this book you will find all of the problems that he proposed for Crux Mathematicorum, including all of his Quickies. His prob-lems covered a very wide range of topics, and show a great deal of insight into what is possible in these areas. The problems are arranged into sets according to topic, and the lightly edited solutions are as published in Crux Mathematicorum (now Crux Mathematicorum with Mathematical Mayhem).

Problems from Murray Klamkin: The Canadian Collection is published by the Mathematical Association of America (MAA) in collaboration with the Canadian Mathematical Society (CMS). It is the first volume in the Canadian Collection.

AMS / MAA PROBLEM BOOKS