The Colorado Mathematical Olympiad: The Third …978-3-319-52861...Forewords to “The Colorado...

The Colorado Mathematical Olympiad:The Third Decade and Further Explorations

Alexander Soifer

The Colorado MathematicalOlympiad: The Third Decade

and Further Explorations

From the Mountains of Coloradoto the Peaks of Mathematics

Forewords by

Branko GrunbaumPeter D. Johnson, Jr.

Alexander SoiferCollege of Letters, Arts, and SciencesUniversity of Colorado at Colorado Springs1420 Austin Bluffs ParkwayColorado Springs, CO [email protected]

ISBN 978-3-319-52859-5 ISBN 978-3-319-52861-8 (eBook)DOI 10.1007/978-3-319-52861-8

Library of Congress Control Number: 2017934206

Mathematics Subject Classification: 2000

© Alexander Soifer 2017This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part ofthe material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmissionor information storage and retrieval, electronic adaptation, computer software, or by similar ordissimilar methodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exemptfrom the relevant protective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in thisbook are believed to be true and accurate at the date of publication. Neither the publisher nor theauthors or the editors give a warranty, express or implied, with respect to the material containedherein or for any errors or omissions that may have been made. The publisher remains neutral withregard to jurisdictional claims in published maps and institutional affiliations.

Cover Illustrations: The illustrations on the front cover, from the upper left clockwise, come fromsolutions of problems 28.5; 21.5; Further Exploration E30; and solution of problem 26.3; allpresented in this book.

Printed on acid-free paper

This Springer imprint is published by Springer NatureThe registered company is Springer International Publishing AGThe registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

mailto:10.1007/978-3-319-52861-8

To all those peoplethroughout the world

who create Olympiadsfor new generations of mathematicians.

Forewords to “The Colorado Mathematical

Olympiad, The Third Decade and Further

Explorations: From the Mountains of

Colorado to the Peaks of Mathematics”

Having written enthusiastic forewords to the first and second install-

ments of Alexander Soifer’s series of books about the ColoradoMathematical Olympiads that covered the first two decades of thesetremendously successful and popular events, it is not hard to antici-pate my feelings about the third installment. I need to admit that I amalmost speechless (and will stop speaking when done with this fore-word) facing the ingenuity and inventiveness demonstrated in theproblems proposed in the third decade of these Olympics. However,equally impressive is the drive and persistence of the originator andliving soul of them. It is hard for me to imagine the enthusiasm andcommitment needed to work singlehandedly on such an endeavorover several decades. True, in the more recent past there was helpfrom various quarters (in contrast to the situation Soifer encounteredin some of the past years)—but still the single-minded drive is morethan admirable.

Any mathematician will derive much pleasure reading about theproblems, but so will students (even middle and high school ones)who are interested in challenging puzzles that do not require any“higher mathematics.” As Soifer repeatedly points out, no advancedknowledge is needed to understand the problems, and only

vii

willingness to think logically and with full attention is needed for thesolutions. The full benefit of the Olympiad and the spirit it fosters isevident by the many alumni who went on to become successful inboth mathematics and other fields.

The text in enlivened by Soifer’s comments and anecdotes, whichgreatly enrich the experience provided by the book. Soifer does nothesitate to be explicit about his opinions regarding ethics in mathe-matics (and in general). In particular, he is deeply wounded by thenaming of a medal (for outstanding mathematical research) of theInternational Mathematical Union after Rolf Nevanlinna. DuringWorld War II Nevanlinna lavishly praised Hitler and was active in

recruitment in Finland of the infamous SS troops.The enthusiasm with which students and their teachers endorse

Soifer’s Olympiads points to a regrettable gap in the curricula ofour schools. Painting with a very broad brush, one can say thatthroughout their education the students are not exposed to any kindof visual geometry of the type exemplified in the problems of theOlympiads, nor are they getting any exposure to combinatorial think-ing. Both of these are often helpful in students’ later education,and life.

As of this writing, Soifer continues in organizing the fourth decadeof the Olympics. One can only wish him (and us) many more years ofsuccess.

Branko GrünbaumDepartment of Mathematics

University of Washington, SeattleNovember 24, 2016

viii Foreword

Here is another gem from Alexander Soifer, the third in a series on theColorado Mathematical Olympiad, founded by Professor Soifer in1984 and recreated by him every year since. As in the previous twovolumes, the meat of the feast is in the problems and solutions foreach Olympiad, but these delights are accompanied and seasoned byan account, for each Olympiad, of who won what, which schools thewinners came from, what grade they were in (the Olympiad hascontestants from grades 6 through 12!), who their teachers were,and whether they had achieved distinction in previous Olympiads.Right, all of this is for the record, like a line of hieroglyphics detailingthe accomplishments of a pharaoh, but the record for each year refersto years past, and even if you skim these sections, paying scantattention, the drama builds: Hannah Alpert places third (out of hun-dreds) one year, second the next, and then, in her senior year. . .secondagain. But in that year she and the fellow that placed ahead of her thatyear and the year before were judged to be so many streets ahead ofeverybody else that no third place was awarded—this had neverhappened before and has never happened since. We find that someyears later, after finishing their undergraduate degrees, both contes-tants wound up in the mathematics Ph.D. program at M.I.T. Yes, thatM.I.T.

And then there are the accounts of difficulties in holding theevent—snow storms about every third year, Alex Soifer in and outof the hospital with a kidney stone around the time of the 2006Olympiad. Another strand that I found to be interesting has to dowith Alex’s indefatigable efforts to get politicians to attend the

ix

Awards Ceremony of the Olympiad. They never come, but many sendletters, every one of which is published in the book—letters from statesenators and the three governors that were in office in Coloradoduring the 10-year span covered by this volume.

Not that the letters are especially interesting to read, but the factthat they were written is of great interest in view of the fact that duringthe first 5 years of the Olympiads, 1984–1988, Professor Soifer had tofight like the dickens to keep the Olympiads alive. A shortsighteddepartment head colluded with misguided administrators at UCCS toarrive at the conclusion that holding an annual math test for grades 6–12 was an unworthy pursuit for a UCCS professor, just not worth the

trouble and expense. It was touch-and-go for a while, but eventuallyAlex was not fired and the Olympiads were not discontinued, thanksto Professor Soifer’s political agility and the presence of UCCSadministrators of a different breed from the ones threatening Olym-piad destruction. What I see in Alex’s pursuit of the attention ofpoliticians and journalists, and his policy of prominently displayingfavorable newspaper coverage and letters from eminent personswhenever possible, is a shrewd and skillful cementing of the statusof the Colorado Olympiad so that it will never be threatened withabolition again, at least while Alexander Soifer is alive andfunctioning.

One consequence of Professor Soifer’s success in this regard is thatthe historical narrative that accompanies the problems and solutionsin this account of the third decade of the Olympiad cannot possibly beas exciting as the corresponding narrative in the accounts of the first20 Olympiads, although some of us still find it quite interesting. But,in compensation, I must tell you that the other two constituents of thebook—the problems and solutions, and the “further explorations”—have, in my opinion, struck a new level.

Alexander Soifer grew up in the Soviet Union where, as in much ofEastern Europe, mathematical exam competitions for secondaryschool students are common, and where the ability to make upsuitable problems for such exams is a prized talent. Alex himselfcredits participation in such competitions for awakening his interestin mathematics, and he has always been an advocate of introducingstudents to mathematics by posing problems—interesting problems,appropriate to their level. If I am not mistaken, his first book wasentitled Mathematics as Problem Solving.

x Foreword

I am of the opinion that if you work at something a long time—say,20 years or more—and really try to get better at it, whatever it is, thenyou will get better at it, even if you were pretty good at it to start with.As I peruse the problems of the third decade of the Colorado Olym-piad, I think I see that this has happened to Alex Soifer. He hasalways, in adulthood, had a great nose for great problems, and hehas always been good at making up problems himself, but now, afterdecades of hunting for Olympiad problems, and struggling to createOlympiad problems, he has become an extraordinary connoisseur andcreator of Olympiad problems. The Olympiad problems were verygood, from the beginning, but in the third decade the problems have

become extraordinarily good. Every brace of five problems is a workof art. The harder individual problems range in quality from brilliantto work-of-genius. I wish that I could find time in my busy schedule towallow in this book for a couple of months. Every time I pick aproblem at random and give it some time—at least half an hour, say—I feel myself getting smarter.

And the same goes for the “Further Explorations” part of the book.Great mathematics and mathematical questions are immersed in asauce of fascinating anecdote and reminiscence. If you could haveonly one book to enjoy while stranded on a desert island, this wouldbe a good choice. If you know a teenager who is even mildlyinterested in mathematics and you wanted to give the teenager a giftof one book, this would be a good choice.

Peter D. Johnson, Jr.Department of Mathematics and Statistics

Auburn UniversityDecember 5, 2016

Foreword xi

The 2011 Forewords for “The Colorado

Mathematical Olympiad and Further

Explorations: From the Mountains of

Colorado to the Peaks of Mathematics”

We live in an age of extreme specialization—in mathematics as well

as in all other sciences, in engineering, in medicine. Hence, to say thatprobably 90% of mathematicians cannot understand 90% of mathe-matics currently published is, most likely, too optimistic. In contrast,even a pessimist would have to agree that at least 90% of the materialin this book is readily accessible to, and understandable by, 90% ofstudents in middle and high schools. However, this does not mean thatthe topics are trivial—they are elementary in the sense that they donot require knowledge of lots of previously studied material, but aresophisticated in requiring attention, concentration, and thinking thatis not fettered by preconceptions. The organization in groups of fiveproblems for each of the “Olympiads,” for which the participantswere allowed four hours, hints at the difficulty of finding completesolutions. I am convinced that most professional mathematicianswould be hard pressed to solve a set of five problems in two hours,or even four.

There are many collections of problems, for “Olympiads” of var-ious levels, as well as problems in a variety of journals. What sets thisbook apart from the “competition” are several aspects that deserve tobe noted.

xiii

• The serenity and enthusiasm with which the problems, and theirsolutions, are presented;

• The absence of prerequisites for understanding the problems andtheir solutions;

• The mixture of geometric and combinatorial ideas that are requiredin almost all cases.

The detailed exposition of the trials and tribulations endured by theauthor, as well as the support he received, shed light on the variety ofinfluences which the administration of a university exerts on thefaculty. As some of the negative actions are very probably a conse-quence of mathophobia, the spirit of this book may cure at least a few

present or future deciders from that affliction.Many mathematicians are certainly able to come up with an inter-

esting elementary problem. But Soifer may be unique in his persis-tence, over the decades, of inventing worthwhile problems, andproviding amusing historical and other comments, all accessible tothe intended pre-college students.

It is my fervent hope that this book will find the wide readership itdeserves, and that its readers will feel motivated to look for enjoy-ment in mathematics.

Branko GrünbaumDepartment of Mathematics

University of Washington, Seattle

xiv Foreword

Here is another wonderful book from Alexander Soifer. This one is amore-than-doubling of an earlier book on the first 10 years of theColorado Mathematical Olympiad, which was founded and nourishedto robust young adulthood by Alexander Soifer.

Like The Mathematical Coloring Book, this book is not so muchmathematical literature as it is literature built around mathematics, ifyou will permit the distinction. Yes, there is plenty of mathematicshere, and of the most delicious kind. In case you were unaware of, orhad forgotten (as I had), the level of skill, nay, art, necessary to posegood olympiad—or Putnam exam-style problems, or the effect thatsuch a problem can have on a young mind, and even on the thoughtsof a jaded sophisticate, then what you have been missing can be foundhere in plenty—at least a year’s supply of great intellectual gustation.If you are a mathematics educator looking for activities for a math

club—your search is over! And with the Further Explorations sec-tions, anyone so inclined could spend a lifetime on the mathematicssprouting from this volume.

But since there will be no shortage of praise for the mathematicaland pedagogical contributions of From the Mountains of Colorado. . ., let me leave that aspect of the work and supply a few words aboutthe historical account that surrounds and binds the mathematicaltrove, and makes a story of it all. The Historical Notes read like awar diary, or an explorer’s letters home: there is a pleasant, mundanerhythm of reportage—who and how many showed up from where,who the sponsors were, which luminaries visited, who won, whattheir prizes were—punctuated by turbulence, events ranging from

xv

A. Soifer’s scolding of a local newspaper for printing only the namesof the top winners, to the difficulties arising from the weather and theshootings at Columbine High School in 1999 (both matters of life anddeath in Colorado), to the inexplicable attempts of university admin-istrators to impede, restructure, banish, or destroy the ColoradoMathematical Olympiad, in 1985, 1986, 2001, and 2003. It is fasci-nating stuff. The very few who have the entrepreneurial spirit toattempt the creation of anything like an Olympiad will be forewarnedand inspired.

The rest of us will be pleasurably horrified and amazed, oursympathies stimulated and our support aroused for the brave ones

who bring new life to the communication of mathematics.

Peter D. Johnson, Jr.Department of Mathematics and Statistics

Auburn University

xvi Foreword

In the common understanding of things, mathematics is dispassionate.This unfortunate notion is reinforced by modern mathematical prose,which gets good marks for logic and poor ones for engagement. Butthe mystery and excitement of mathematical discovery cannot bedenied. These qualities overflow all preset boundaries.

On July 10, 1796, Gauss wrote in his diary

EϒΡΗKΑ! num ¼ Δþ Δþ Δ

He had discovered a proof that every positive integer is the sum ofthree triangular numbers 0, 1, 3, 6, 10, . . . n nþ 1ð Þ=2, . . .f g. Thisresult was something special. It was right to celebrate the momentwith an exclamation of Eureka!

In 1926, an intriguing conjecture was making the rounds ofEuropean universities.

If the set of positive integers is partitioned into two classes, thenat least one of the classes contains an n-term arithmetic progres-sion, no matter how large n is taken to be.

The conjecture had been formulated by the Dutch mathematicianP. J. H. Baudet, who told it to his friend and mentor Frederik Schuh.B. L. van der Waerden learned the problem in Schuh’s seminar at theUniversity of Amsterdam. While in Hamburg, van der Waerden toldthe conjecture to Emil Artin and Otto Schreier as the three hadlunch. After lunch, they adjourned to Artin’s office at the Universityof Hamburg to try to find a proof. They were successful, and the

xvii

result, now known as van der Waerden’s Theorem, is one of the ThreePearls of Number Theory in Khinchine’s book by that name.The story does not end there. In 1971, van der Waerden published aremarkable paper entitled How the Proof of Baudet’s Conjecture wasFound.1 In it, he describes how the three mathematicians searched fora proof by drawing diagrams on the blackboard to represent theclasses, and how each mathematician had Einf€alle (sudden ideas)that were crucial to the proof. In this account, the reader is a fourthperson in Artin’s office, observing with each Einfall the rising antic-ipation that the proof is going to work. Even though unspoken, eachof the three must have had a “Eureka moment” when success was

assured.From Colorado Mountains to the Peaks of Mathematics presents

the 20-year history of the Colorado Mathematical Olympiad. It issymbolic that this Olympiad is held in Colorado. Colorado is knownfor its beauty and spaciousness. In the book there is plenty of spacefor mathematics. There are wonderful problems with ingenious solu-tions, taken from geometry, combinatorics, number theory, and otherareas. But there is much more. There is space to meet the participants,hear their candid comments, learn of their talents, mathematical andotherwise, and in some cases to follow their paths as professionals.There is space for poetry and references to the arts. There is space fora full story of the competition—its dreams and rewards, hard workand conflict. There is space for the author to comment on matters ofgeneral concern. One such comment expresses regret at the limita-tions of currently accepted mathematical prose.

In my historical-mathematical research for The Mathematical Col-oring Book, I read a good number of nineteenth-century Victorianmathematical papers. Clearly, the precision and rigor of mathemat-ical prose has improved since then, but something charming waslost—perhaps, we lost the “taste of time” in our demand for an“objective,” impersonal writing, enforced by journal editors andmany publishers. I decided to give a historical taste to my Olym-pians, and show them that behind Victorian clothing we can findthe pumping heart of the Olympiad spirit. [p. 297]

1Studies in Pure Mathematics (Presented to Richard Rado), L. Mirsky, ed., Academic Press,

London, 1971, pp 251–260.

xviii Foreword

Like Gauss, Alexander Soifer would not hesitate to inject Eureka!at the right moment. Like van der Waerden, he can transform adispassionate exercise in logic into a compelling account of suddeninsights and ultimate triumph.

Cecil RousseauProfessor Emeritus

University of MemphisChair, USA Mathematical Olympiad Committee

Foreword xix

The 1994 Forewords for “Colorado

Mathematical Olympiad: The First

10 Years and Further Explorations”

Love! Passion! Intrigue! Suspense! Who would believe that the

history of a mathematics competition could accurately be describedby words that more typically appear on the back of a popular novel?After all, mathematics is dull; history is dull; school is dull. Isn’t thatthe conventional wisdom?

In describing the history of the Colorado Mathematical Olympiad,Alexander Soifer records the comments of a mathematics teacher whoanonymously supported the Olympiad in each of its first 10 years.When asked why, this unselfish teacher responded “I love my profes-sion. This is my way to give something back to it.” Alexander alsoloves his profession. He is passionate about his profession. And heworks hard to give something back.

The Colorado Mathematical Olympiad is just one way Alexanderdemonstrates his love for mathematics, his love for teaching, his lovefor passing on the incredible joy of discovery. And as you read thehistory of the Olympiad, you cannot help but be taken up yourselfwith his passion.

But where there is passion, there is frequently intrigue. Here itinvolves the efforts of school administrators and others to help—or tohinder—the success of the Olympiad. But Alexander acknowledges

xxi

that we all must have many friends to help us on the journey tosuccess. And the Olympiad has had many friends, as Alexander socarefully and thankfully records.

One of the great results of the Olympiad is the demonstration thatreal mathematics can be exciting and suspenseful. But the Olympiadalso demonstrates the essence of mathematical research, or whatmathematicians really do as they move from problem to example togeneralization to deeper results to new problem. And in doing so itprovides an invaluable lesson to the hundreds of students who partic-ipate each year.

It is appropriate on an anniversary to look back and take stock. It is

also appropriate to look forward. This book does both, for the Colo-rado Mathematical Olympiad is alive and well, thanks to its manyardent supporters. And for that we can all rejoice.

Philip L. EngelPresident, CNA Insurance Companies

Chairman of the Board, MATHCOUNTS FoundationMarch 28, 1994, Chicago, Illinois

xxii Foreword

The author started the Colorado Olympiad in 1984, 10 years ago, andit was a complete success and it is continuing. Several of the winnershave already got their PhDs in Mathematics and Computer Science.

The problems are discussed with their solutions in great detail.A delightful feature of the book is that in the second part more relatedproblems are discussed. Some of them are still unsolved; e.g., theproblem of the chromatic number of the plane—two points of theplane are joined if their distance is 1—what is the chromatic numberof this graph? It is known that it is between 4 and 7. I would guess thatit is greater than 4 but I have no further guess. Just today (March8, 1994) Moshe Rosenfeld asked me—join two points of the plane iftheir distance is an odd integer—is the chromatic number of thisgraph finite? He proved that if four points are given, the distancescannot all be odd integers.

The author states an unsolved problem of his and offers a prize of$100 for it. For a convex figure F in the plane, S(F) denotes theminimal positive integer n, such that among any n points inside or onthe boundary of F there are three points that form a triangle of area14Fj j or less, where |F| is the area of F. Since for any convex figure F,

S(F)¼ 5 or 6, it is natural to ask for a classification of all convexfigures F, such that S(F)¼ 6.

xxiii

I warmly recommend this book to all who are interested in difficultelementary problems.

Paul ErdosMember of the Hungarian Academy of Sciences

Honorary Member of the NationalAcademy of Sciences of the USA

Boca Raton, FloridaMarch 8, 1994

xxiv Foreword

Alexander Soifer, who founded and still runs the famous ColoradoMathematics Olympiad, is one of the world’s top creators of signif-icant problems and conjectures. His latest book covers the Olym-piad’s first 10 years, followed by additional questions that flow fromOlympiad problems.

The book is a gold mine of brilliant reasoning with special empha-sis on the power and beauty of coloring proofs. Stronglyrecommended to both serious and recreational mathematicians onall levels of expertise.

Martin GardnerHendersonville, North Carolina

March 10, 1994

xxv

Many of us wish we could contribute to making mathematics moreattractive and interesting to young people. But few among profes-sional mathematicians find the time and energy to actually do much inthis direction. Even fewer are enterprising enough to start acompletely new project and continue carrying it out for many years,making it succeed against overwhelming odds. This book is anaccount of such a rare endeavor. It details one person’s single-mindedand unwavering effort to organize a mathematics contest meant forand accessible to high school students. Professor Soifer managed tosecure the help of many individuals and organizations; surprisingly,he also had to overcome serious difficulties which should not havebeen expected and which should not have arisen.

The book is interesting in many ways. It presents the history of thestruggle to organize the yearly “Colorado Mathematical Olympiad”;

this should help others who are thinking of organizing similar pro-jects. It details many attractive mathematical questions, of varyingdegrees of difficulty, together with the background for many of themand with well-explained solutions, in a manner that students as well asthose who try to coach them will find helpful. Finally, the “FurtherExplorations” make it clear to the reader that each of these ques-tions—like all of mathematics—can be used as a stepping stone toother investigations and insights.

I finished reading the book in one sitting—I just could not put itdown. Professor Soifer has indebted us all by first making the effort

xxvii

to organize the Colorado Mathematical Olympiads, and then makingthe additional effort to tell us about it in such an engaging anduseful way.

Branko GrünbaumUniversity of Washington

Seattle, WashingtonMarch 18, 1994

xxviii Foreword

If one wants to learn about the problems given at the 1981 Interna-tional Mathematical Olympiad, or to find a statistical summary of theresults of that competition, the required information is contained inMurray Klamkin’s book International Mathematical Olympiads1979–1985. To find out about the members of the 1981 USA team(Benjamin Fisher, David Yuen, Gregg Patruno, Noam Elkies, JeremyPrimer, Richard Stong, James Roche, and Brian Hunt) and what theyhave accomplished in the intervening years, one can read the bookletWho’s Who of U.S.A. Mathematical Olympiad Participants 1972–1986 by Nura Turner. For a view from behind the scenes at the 1981IMO, there is the interesting article by Al Willcox, “Inside the IMO,”in the September–October, 1981, issue of Focus. News accounts ofthe IMO can be found in the Time and Newsweek as well as majornewspapers. However, even if one is willing to seek out these various

sources, it is hard to get a full picture of such a MathematicalOlympiad, for it is much more than a collection of problems and astatistical summary of results. Its full story must be told in terms ofdreams, conflicts, frustration, celebration, and joy.

Now thanks to Alexander Soifer, there is a book about the ColoradoMathematical Olympiad that gives more than just the problems, theirsolutions, and statistical information about the results of the compe-tition. It tells the story of this competition in direct, human terms.Beginning with Soifer’s own experience as a student in Moscow,Colorado Mathematical Olympiad describes the genesis of the math-ematical competition he has created and gives a picture of the workrequired to gain support for such a project. It mentions participants byname and tells of some of their accomplishments. It acknowledges

xxix

those who have contributed problems and it reveals interestingconnections between the contest problems and mathematicalresearch. Of course, it has a collection of mathematical problemsand solutions, very beautiful ones. Some of the problems are fromthe mathematical folklore, while others are striking original contri-butions of Soifer and some of his colleagues. Here’s one of myfavorites, a problem contributed by Paul Zeitz.

Twenty-three people of positive integral weight decide to playfootball. They select one person as referee and then split up intotwo 11-person teams of equal total weight. It turns out that nomatter who is chosen referee this can always be done. Prove that all

23 people have the same weight.

The problems alone would make this book rewarding to read. ButColorado Mathematical Olympiad has more than attractive mathe-matical problems. It has a compelling story involving the lives ofthose who have been part of this competition.

Cecil RousseauMemphis State University

Coach, U.S.A. team for the InternationalMathematical Olympiad

Memphis, TennesseeApril 1, 1994

xxx Foreword

Contents

Forewords to “The Colorado Mathematical Olympiad,The Third Decade and Further Explorations: From theMountains of Colorado to the Peaks of Mathematics” . vii

Foreword by Branko Grünbaum . . . . . . . . . . . . viiForeword by Peter D. Johnson, Jr. . . . . . . . . . . . ix

The 2011 Forewords for “The Colorado MathematicalOlympiad and Further Explorations: From theMountains of Colorado to the Peaks of Mathematics” . xiii

Foreword by Branko Grünbaum . . . . . . . . . . . . xiiiForeword by Peter D. Johnson, Jr. . . . . . . . . . . . xvForeword by Cecil Rousseau . . . . . . . . . . . . . . xvii

The 1994 Forewords for “Colorado MathematicalOlympiad: The First 10 Years and FurtherExplorations” . . . . . . . . . . . . . . . . . . . . . . . . . xxi

Foreword by Philip L. Engel . . . . . . . . . . . . . . xxiForeword by Paul Erdos . . . . . . . . . . . . . . . . . xxiiiForeword by Martin Gardner . . . . . . . . . . . . . . xxvForeword by Branko Grünbaum . . . . . . . . . . . . xxviiForeword by Cecil Rousseau . . . . . . . . . . . . . . xxix

xxxi

Greetings to the Reader—2017 . . . . . . . . . . . . . . . xxxvii

Greetings to the Reader—2011 . . . . . . . . . . . . . . . xliii

Greetings to the Reader—1994 . . . . . . . . . . . . . . . xlix

Part I. The Third Decade

Twenty-First Colorado Mathematical Olympiad

April 16, 2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Historical Notes 21 . . . . . . . . . . . . . . . . . . . . . . 3Problems 21 . . . . . . . . . . . . . . . . . . . . . . . . . . 8Solutions 21 . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Twenty-Second Colorado Mathematical Olympiad


Twenty-Third Colorado Mathematical Olympiad


Twenty-Fourth Colorado Mathematical Olympiad


Twenty-Fifth Colorado Mathematical Olympiad

April 18, 2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Historical Notes 25 . . . . . . . . . . . . . . . . . . . . . . . 71Problems 25 . . . . . . . . . . . . . . . . . . . . . . . . . . 78Solutions 25 . . . . . . . . . . . . . . . . . . . . . . . . . . 79

xxxii Contents

Twenty-Sixth Colorado Mathematical Olympiad


Twenty-Seventh Colorado Mathematical Olympiad


Twenty-Eighth Colorado Mathematical Olympiad

April 22, 2011 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111Historical Notes 28 . . . . . . . . . . . . . . . . . . . . . . . 111Is Mathematics an Art? . . . . . . . . . . . . . . . . . . . . 115Problems 28 . . . . . . . . . . . . . . . . . . . . . . . . . . 117Solutions 28 . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Twenty-Ninth Colorado Mathematical Olympiad


Thirtieth Colorado Mathematical Olympiad

April 26, 2013 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135Historical Notes 30 . . . . . . . . . . . . . . . . . . . . . . . 135

Problems 30 . . . . . . . . . . . . . . . . . . . . . . . . . . 143Solutions 30 . . . . . . . . . . . . . . . . . . . . . . . . . . 144

A Round Table Discussion of the Olympiad, or LookingBack from a 30-Year Perspective . . . . . . . . . . . . . 153

Contents xxxiii

Part II. Further Explorations of the Third Decade

Introduction to Part II . . . . . . . . . . . . . . . . . . . . . . 169

E21. Cover-Up with John Conway, Mitya Karabash,and Ron Graham . . . . . . . . . . . . . . . . . . . . . . 171

E22. Deep Roots of Uniqueness . . . . . . . . . . . . . . . . . 183

E23: More about Love and Death . . . . . . . . . . . . . . . . 185

E24: One Amazing Problem and Its Connectionsto Everything—A Conversationin Three Movements . . . . . . . . . . . . . . . . . . . . 189

E25: The Story of One Old Erdos Problem . . . . . . . . . . 199

E26: Mark Heim’s Proof . . . . . . . . . . . . . . . . . . . . . 203

E27: Coloring Integers—Entertainmentof Mathematical Kind . . . . . . . . . . . . . . . . . . . . 205

E28: The Erdos Number and Hamiltonian Mysteries . . . . 213

E29: One Old Erdos–Turan Problem . . . . . . . . . . . . . . 217

E30: Birth of a Problem: The Story of Creationin Seven Stages . . . . . . . . . . . . . . . . . . . . . . . . 221

Part III. Olympic Reminiscences in Four Movements

Movement 1. The Colorado Mathematical OlympiadIs Mathematics; It Is Sport; It Is Art.And It Is Also Community,by Matthew Kahle . . . . . . . . . . . . . . . . . 231

Movement 2. I’ve Begun Paying Off My Debtwith New Kids, by Aaron Parsons . . . . . . . 235

Movement 3: Aesthetic of Personal Mastery,by Hannah Alpert . . . . . . . . . . . . . . . . . 239

xxxiv Contents

Movement 4. Colorado Mathematical Olympiad:Reminiscences by Robert Ewell . . . . . . . . . 243

Farewell to the Reader . . . . . . . . . . . . . . . . . . . . . . 249

References . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

Index of Names . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

Index of Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

Contents xxxv

Greetings to the Reader—2017

As far as the laws of mathematics refer to reality,they are not certain, and as far as they are certain,they do not refer to reality.

—Albert Einstein

The world is a dangerous place to live; notbecause of the people who are evil, but becauseof the people who don’t do anything about it.

—Albert Einstein

The Colorado Mathematical Olympiad has survived for over30 years—long live the Olympiad! It has become a part of our culturallife in Colorado. During the first 30 years, 18,000 students partici-pated in the Olympiad. They wrote 89,000 essays and were awardedover $317,000 in prizes. The Olympiad is a unique joint effort ofschool districts, schools, institutions of higher education, businesscommunity, and local and state governments.

What new can I say here after I gave a lot of thought to the previoustwo 2011 and 1994 forewords reproduced below? The author—I—changed over the past 30 years, as did my views of priorities in life,and consequently my priorities in science and education.

Most of my life I have been a student of Beauty in all her mani-festations, especially nature and the arts. Mathematics is one of the

xxxvii

arts for me. However, my 20-year long 1995–2015 work on the bookThe Scholar and the State: In Search of Van der Warden [Soi10] haschanged me. I now view the highest priority of instruction andresearch in upholding high moral principles. This is important!Many of my colleagues believe that mathematics we create is allthat matters, Mathematik €uber Alles, mathematics above all moralconcerns. In my opinion, there is no good science or good art unless itis built on the foundation of high ethical principles. Often publishersof mathematical books and journals downplay and sometimes evendisregard ethical principles. This is so shortsighted! We have seen inhistory time and again how evil the usage of science could be if it is

not built on high moral foundation. Atrocities of Nazi Germany aloneprovide countless examples of how science, technology, and even artcan be used for ill deeds. Mentioned above my book [Soi10] isdedicated precisely to moral dilemmas of a scholar in the ThirdReich and in the world of today, the world in which Russia invadesand wages wars against its sovereign neighbors Moldova, Georgia,and Ukraine. In order for creative work to be good, it must also servethe good. It ought to be humane. It has to be grounded in morality,empathy, compassion, and kindness. The Great Russian poet Alexan-der Pushkin (1799–1837) poetically expressed this. Let me translatehis lines for you:

And people will be pleased with me for years to come,For I awakened kindness with my lyre,For in my cruel age I Freedom praised and sangAnd urged I mercy for the fallen people.

Those fluent in Russian will appreciate the original verses:

И дoлгo буду тeм любeзeн я нapoду,Чтo чувcтвa дoбpыe я лиpoй пpoбуждaл,Чтo в мoй жecтoкий вeк вoccлaвил я CвoбoдуИ милocть к пaдшим пpизывaл.

I hope that you, my young colleagues, would accept the baton ofmercy and humanity and by your creative work will contribute to thehigh culture of our small endangered planet.

To my disbelief, I encountered not only enthusiastic support of myaddressing ethics of a scholar, but also rare yet vigorous opposition.Are mathematicians entitled to a life in Ivory Tower and disregard the

xxxviii Preface

outside world? This must have been a reason the Executive Commit-tee of the International Mathematics Union (IMU) established in 1981The Rolf Nevanlinna Prize that includes a gold medal with the profileof this fine mathematician, who during the World War II served as theChair of Recruitment Committee of the SS troops in Finland, whopraised Hitler (!) as liberator of Europe. No doubt you know that theSS troops were responsible for the majority of crimes against human-ity as was determined in Nuremberg Trials and other post-World WarII tribunals.

I urged the name change of this prize and the medal in my book[Soi10], in my July 23, 2016, public address to the General Assembly

of the International Commission on Mathematics Instruction (ICMI),and in my letter to the Executive Committee of IMU written on behalfof the Executive Committee of the World Federation of NationalMathematics Competitions (of which I am President). IMU PresidentShigefumi Moro promised me to consider this request at the nextmeeting of the IMU Executive Committee in April 2017. They maydecide not to change the name, but we have done all we can, and thatcounts for something. The 1986 Nobel Peace Laureate Elie Wieselcalled upon us to stand up and be counted:

There may be times when we are powerless to prevent injustice, butthere must never be a time when we fail to protest.

I agree with the great Dutch mathematician Luitzen Egbertus JanBrouwer, who wrote:

It is my opinion that the tiniest moral matter is more important thanall of science, and that one can only maintain the moral quality ofthe world by standing up to any immoral project.

If we, mathematicians of today, took a greater care of our ethics,we would not have lost such a genius mathematician as GrigoryPerelman. Following his great achievements of proving the PoincareConjecture and the Geometrization Conjecture, Grigory walked awayfrom mathematics. He did not want to be “a poster boy” for thescience where the majority tolerates immoral acts of the minority ofits members.

I have always believed that the main goal of mathematics instruc-tion ought to be demonstrating what mathematics is and what math-ematicians do. Technically speaking, we cannot teach science—any

Preface xxxix

science, and any art. We can only create atmosphere for our studentswhere they learn our subject by doing it. As Einstein said,

The only source of knowledge is experience.

Unlike almost any other competition of mathematical kind, ourOlympiad builds bridges between problems of Olympiads and prob-lems of “real” mathematics. These bridges show that the differencebetween the two is quantitative and not qualitative. The same goals ofbeauty, elegance, and surprises for our intuition guide both. And thedifference is mere quantitative: our Olympiad’s problems 3, 4, and5 may require 4–5 hours to solve whereas problems of “real” math-

ematics may require 5000 hours, and sometimes more. As a conse-quence, all my Colorado Mathematical Olympiad books, includingthe present one, feature the bridges between Olympiad problems andresearch mathematics, which I call Further Explorations. This bookcontains ten such Explorations, essays that vividly demonstrate thisconnection, the two-way bridge between the problems we use in ourOlympiad and problems mathematicians ponder in their research,including open problems of mathematics. Yes, two-way, for someof the Olympiad problems pave the way for new mathematicalresearch.

Olympiads inevitably have an element of sport. I try to reduce it byoffering five problems of increasing difficulty that require practicallyno topical knowledge, and 4 hours to solve them and write completeessay-type solutions. The influence of Russian Olympiads of all levelscan easily be noticed. However, there is one more essential element inthe Colorado Mathematical Olympiad. I wish to allow young Olym-pians to compete not only with each other but also with the field ofMathematics. I wish to stop discrimination of high school mathema-ticians based on their tender age, and inspire them to engage inresearch. To this end, our more difficult problems often open horizonsand allow for further explorations, leading young Olympians to theforefront of mathematics.

Authors of the problems are listed next to the problems’ title.However, what could I do when I slightly changed a problem thatcame to me from the (Russian) mathematical folklore? Everyone isthe author, and no one is at the same time. In those cases I did not listany authorship. Every year, the problems have been selected andedited by the Problem Committee, which included Col. Dr. Robert

xl Preface

“Bob” Ewell and me. During some years, the Committee alsoincluded Gary Miller.

I thank the historic Springer for inviting this book—my eighthbook with Springer—in its historic publishing house, founded in1842. No words can fully express the depth of my gratitude to BrankoGrünbaum and Peter D. Johnson, Jr., who were the first readers of thisand all my previous books. Their forewords and suggestions havealways been grounded in their great intellectual capacity, mathe-matical brilliance, perfect aesthetic taste, and unwavering moralcompass.

I thank all the numerous people who joined me in this major

undertaking and made the Olympiad possible, judges, proctors, staffmembers of the University of Colorado Colorado Springs, sponsors,and above all Olympians and their families. A special gratitude goesto Bob Ewell, an Olympiad’s senior judge for a quarter a century, whotranslated my sketches into computer-aided illustrations you see inthis book.

I thank all my mathematics teachers from the grade school to mygrown-up stage in life: Matilda I. Koroleva, Klara A. Dimonstein,Nikolai N. Konstantinov, Tatiana N. Fideli, Ivan V. Morozkin, YuriF. Mett, Leonid Ya. Kulikov, and Paul Erdos. I am grateful to myparents, artist Yuri A. Soifer and actress Frieda M. Hoffman Soifer,for moral guidance and intellectual environment they provided.I thank my kids—Mark, Julia, Isabelle, and Leon—for support andinspiration. Once again, I dedicate my Olympiad book to all thosepeople around the world who create mathematical Olympiads for newgenerations of mathematicians.

At the Colorado Mathematical Olympiad, we have been oftenasked a natural question: how does one create a Mathematical Olym-piad and how does it work? This and other related questions areclarified by the University of Colorado, which produced the 2014film “Thirtieth Colorado Mathematical Olympiad—30 Years ofExcellence.” You can find it on the Olympiad’s homepage http://olympiad.uccs.edu/ and on the YouTube.

I started this foreword with two epigraphs from Albert Einstein. Incase you do not see their relevance to this book, let me clarify. Thefirst one, in a sense, implies that mathematics is not a natural science,it is an art. The second quotation calls on us all to stand up and becounted in the pursuit of high professional ethics. I concur.

Preface xli

http://olympiad.uccs.edu/

http://olympiad.uccs.edu/


Beauty is an instance which plainly shows thatculture is not simply utilitarian in its aims, for thelack of beauty is a thing we cannot tolerate incivilization.

—Sigmund Feud, 1930Civilization and Its Discontents [F]

Talent is not performance; arms and legs are nodance.

—Hugo von Hofmannsthal, 1922Buch der Freunde [Ho]

Imagination is more important than knowledge.For knowledge is limited to all we now know andunderstand, while imagination embraces theentire world, and all there ever will be to knowand understand.

—Albert Einstein

I have been often asked: what are Mathematical Olympiads for? Dothey predict who will go far in mathematics and who will not? In fact,today, in 2010, in Russia, Olympiads are officially used as predictorsof success, for those who placed high enough in important enough

xliii

Olympiads get admitted to important enough universities! Havingspent a lifetime in Olympiads of all levels, from participant to orga-nizer, I have certainly given these kinds of questions a lot of thought.

There is no way to do well by accident, by luck in an Olympiad(i.e., essay-type competition, requiring presentation of complete solu-tions). Therefore, those who even sometimes have done well inOlympiads undoubtedly have talent. Does it mean they will succeedin mathematics? As the great Austrian writer Hugo von Hofmannsthalput it in 1922, Talent is not performance; arms and legs are no dance.With talent, it still takes work, hard work to succeed. I would saytalent imposes an obligation on its owner, a duty not to waste the

talent. I must add, nothing is a guarantee of success; life interferes andthrows barriers on our way. It is critical to know in your gut that inorder to succeed no reason for failure should be acceptable.

The inverse here is not true. Those who have not done well inOlympiads do not necessarily lack talent. They may be late bloo-mers—Einstein comes to mind. Their talent may be in another field oractivity. I believe there are no talentless people—there are peoplewho have not identified their talent and have not developed it.

In addition to allowing talents shine, Olympiads introduce young-sters to the kind of mathematics they may have not seen in school. Itwas certainly a case with me. Olympiads showed me the existence ofmathematics that justifies such adjectives as beautiful, elegant,humorous, defying intuition. Olympiad mathematics inspires andrecruits; it passes the baton to new generation of mathematicians.

And one more thing, which apparently I said long ago (see mynewspaper interview in Historical Notes 14). The students who dowell in Olympiads have freedom of thought. They can look at usualthings in a new way, like great painters and poets do. Olympiadsprovide students with an opportunity to express and appreciate thiscreative freedom.

The Colorado Mathematical Olympiad has survived for over twodecades. Happy anniversary to the Olympiad and all people whomade it possible! The Olympiad has become part of life for manystudents, parents, and schools of Colorado. I know that the Olympiadis approaching when I receive e-mails from eagerly awaiting partic-ipants, their parents, and teachers.

It was reasonable to limit participation to students of the State ofColorado. However, I could not refuse guest participation, and we

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have had Olympians traveling from Long Island, New York, Kansas,and even the national team of the Philippines. For a number of years,winners of the Mobile, Alabama mathematics competition have beenreceiving as a prize a visit to compete in the Colorado MathematicalOlympiad.

I have just now noticed that in preparing for Springer new(expanded) editions of my books Mathematics as Problem Solving,How Does One Cut a Triangle?, and Geometric Etudes in Combina-torial Mathematics, I preserved original books within new editionsand added new parts within new editions. I could not alter thewholeness of the first editions because as works of art, these original

books had to be preserved unaltered. This book dramatically expandsupon the 1994 original Colorado Mathematical Olympiad: The First10 Years and Further Explorations. But the original is preserved as“Book I” (I have added some historical sketches, for after The Math-ematical Coloring Book my writing style has changed), followedby the brand new “Book II” and the new “Part V. Winners Speak:Reminiscences in Eight Parts.”

The original Book I presents the history of the first 10 years of theOlympiad. Book I also presents all 51 problems of the first 10 years ofthe Olympiad with their solutions. History, Problems, and Solutionssections are organized by the year of the Olympiad. Problem numberi.j indicates problem j of the i-th Olympiad. After Historical Notes,Problems, and Solutions come Further Explorations, a unique featureof this book, not found in any of the numerous books reportingRussian, American, Chinese, International, and other Olympiads.Each Exploration takes off from one or more Olympiad problemspresented in this book and builds a bridge to the forefront of mathe-matics, in some explorations to open problems of mathematics. Thisis the feature Paul Erdos liked the most, when he decided to write aforeword for the 1994 edition of the book.

The new Book II presents the history and problems of the seconddecade. I can now see how the problems of the Olympiad havematured over the years. Book II then offers ten new Further Explo-rations, for the total of 20 bridges between Olympiad problems andproblems of real mathematics. I mean here “real” in the sense of realmathematicians working on these kinds of problems, riding thesekinds of trains of thought—and not in a sense of “real” life. In 1921Albert Einstein addressed the correlation between mathematics and

Preface xlv

reality in a most convincing way: As far as the propositions ofmathematics refer to reality, they are not certain; and as far as theyare certain, they do not refer to reality.

Books I and II, each consisting of two parts, are followed by Part V,Winners Speak: Reminiscences in Eight Parts. In this part, severalwinners of the Olympiad evaluate the role of the Olympiad in theirlives and describe their young professional careers, life after theOlympiad.

First of all I thank the many thousands of young mathematicians,Colorado Olympians, for without them, their dedication to the Olym-piad, their efforts, and their demonstrated brilliance all our hard work

would make no sense.I am grateful to my Dean of Letters, Arts and Sciences Tom

Christensen and my Chancellor Pam Shockley-Zalaback forestablishing the Admission Window for the Olympiad’s medalists,the window equal to that for the USA Olympic sportsmen; and toPam for proposing and funding Chancellor’s Scholarships for Olym-piad’s Medalists. I thank the two of them and my chair TomWynn forunderstanding and appreciating the intensity of my ongoing work onthe Olympiad for the past 27+ years.

I thank all those who volunteered their time and talent to serve asjudges and proctors of the Olympiad, especially those judges whohave served the longest: Jerry Klemm, Gary Miller, Bob Ewell, ShaneHolloway, and Matt Kahle. I am grateful to the Olympiad’s managerswho gave the long months every year to organizing the Olympiad andits registration: Andreanna Romero, Kathy Griffith, and MargieTeals-Davis. I thank people of Physical Plant, Media Center, andthe University Center, and first of all Dave Schnabel, ChristianHowells, Rob Doherty, Mark Hallahan, Mark Bell, and Jeff Davis.

The Olympiad has been made possible by dedicated sponsors.I thank them all, and first of all the longest-term sponsorsDr. Stephen Wolfram and Wolfram Research; CASIO; Texas Instru-ments; Colorado Springs School Districts 11, 20, 2, and 3; RangelyHigh School; Chancellor’s and Vice Chancellors’ Offices; and theBookstore of UCCS. I am infinitely indebted to Greg Hoffman, whosefive companies all became major contributors most definitely due toGreg’s trust in and loyalty to the Olympiad.

Every year Colorado Governors Roy Romer, Bill Owens, andBill Ritter, as well as Senator and later Director of Labor Jeffrey

xlvi Preface

M. Welles, have written congratulatory letters to the winners of theOlympiad. Most frequent speakers at the Award Presentation Cere-monies were Chancellors Dwayne Nuzum and Pamela Shockley-Zalaback, Deans James Null and Tom Christensen, sponsor GregHoffman, Senator Jeffrey M. Welles, and Deputy SuperintendantsMaggie Lopes of Air Academy District 12 and Mary Thurman ofColorado Springs District 11. My gratitude goes to all of them and allother speakers of the Award Presentations.

I thank the long-term senior judge of the Olympiad, Dr. Col. RobertEwell, for translating some of my hand-drawn illustrations into sharpcomputer-aided designs.

My late parents, the artist Yuri Soifer and the actress FriedaGofman, gave me life and filled it with art. They rose to the occasion,recognized my enthusiasm toward mathematics, and respected myswitch from the art of music to the art of mathematics. They two andmy kids Mark, Julia, Isabelle, and Leon were my love and inspiration.They all participated in the Olympiad and some won awards. Myveteran judges and I will never forget how for years tiny beautifulIsabelle in a Victorian style dress passed the Olympiad’s buttons tothe winners. I owe you so much!

I admire high talents and professionalism, great taste, and kindattention of the first readers of this manuscript, who are also theauthors of the book’s forewords: Philip L. Engel, Branko Grünbaum,Peter D. Johnson Jr., and Cecil Rousseau. Thank you so very much!

I thank my Springer editor Elizabeth Loew for a constant supportand good cheer, and Ann Kostant for inviting this and my other sevenbooks to the historic Springer.

The Olympiad weathered bad weather postponements and admin-istrators impersonating barricades on the road. The Olympiad alsoencountered enlightened administrators and loyal volunteers. It isalive and well. I may meet with you again soon on the pages of thefuture book inspired by the third decade of the Olympiad!

Preface xlvii


Mathematics, rightly viewed, possesses not onlytruth, but supreme beauty. . . capable of a sternperfection such as only the greatest art can show.

—Bertrand Russell

It has been proved by my own experience thatevery problem carries within itself its ownsolution, a solution to be reached by the intenseinner concentration of a severe devotion to truth.

—Frank Lloyd Wright

The Colorado Mathematical Olympiad has survived for a decade.Happy anniversary to the Olympiad and all people who made itpossible!

What is the Olympiad? Where does one get its problems? How tosolve them? These questions came through the years from everycorner of Colorado and the world. This is the right time to reply: wehave accumulated enough of striking history, intriguing problems,and surprising solutions, and we have not forgotten too much yet.

Part I, The First 10 Years, is my reply to these questions. Itdescribes how and why the Olympiad started and gives a fairlydetailed history of every year. Many, although inevitably far from

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all, people who made the Olympiad possible are recognized: spon-sors, problem creators, judges, proctors, and, of course, contestants.

Some people who tried to make the Olympiad impossible arementioned as well. The history of human events is never a bed ofroses. Future organizers of Olympiads need to know that they willface support and understanding from some and indifference and evenopposition from others.

Part I also presents all 51 problems of the first 10 years of theOlympiad with their solutions. Two or even three solutions arepresented when as many different and beautiful solutions have beenfound. Historical notes, Problems, and Solutions sections are orga-

nized by the Olympiads. Accordingly, problem i.j stands for problemj of the i-th Olympiad.

I did enjoy reliving the 10 years of the Olympiad and revisiting allits problems (nearly all these solutions were written for the first timefor this book). Yet, in working on this book my real inspiration camewhen the idea for Part II: Further Explorations occurred to me. This iswhen I began to feel that I was writing a mathematical book.

Part II in its ten essays demonstrates what happens in a mathemat-ical exploration when a problem at hand is solved. Each essay takesan Olympiad problem (or two, or three of them) and shows how itssolution gives birth to deeper, more exciting, and more generalproblems. Some of these second generation problems are open (i.e.,not solved by anyone!). Others are solved in this book. Severalproblems are left unsolved, even when I know beautiful solutions,to preserve for the reader the pleasure of discovering a solution on hisown. In some essays the reader is led to the third generation problems.Several open problems carry a prize for their first solutions. Forexample, this is the first time that I am offering $100 for the firstsolution of problem E5.8 (this number refers to the eighth problem ofchapter E5 of Further Explorations).

To the best of my knowledge, this is the first Olympiad problembook with a whole special part that bridges problems of mathematicalOlympiads with open problems of mathematics. Yet, it is not a totalsurprise that I am able to offer ten mini-models of mathematicalresearch that originate from Olympiad problems. The famousRussian mathematician Boris N. Delone once said, as AndreiN. Kolmogorov recalls in his introduction to [GT], that in fact, “amajor scientific discovery differs from a good Olympiad problem

l Preface

only by the fact that a solution of an Olympiad problem requires5 hours whereas obtaining a serious scientific result requires 5,000hours.”

There is one more reason why Further Explorations is a key part ofthis book. Olympiads offer high school students an exciting additionand alternative to school mathematics. They show youngsters thebeauty, elegance, and surprises of mathematics. Olympiads celebrateachievements of young mathematicians in their competition witheach other. Yet, I am not a supporter of such competitions foruniversity students. University is a time to compete with the field,not with each other!

I wish to thank here my junior high and high school mathematicsteachers, most of whom are not with us any more: KlaraA. Dimanstein, Tatiana N. Fideli, Boris V. Morozkin, and YuriF. Mett. I just wanted to be like them! I am grateful to NikolaiN. Konstantinov for his fabulous mathematics club that I attendedas an eighth grader, and to the organizers of the Moscow UniversityMathematical Olympiad: they convinced me to become a mathema-tician. I am grateful to the Russian Isaak M. Yaglom and the Amer-ican Martin Gardner for their books that sparked my early interest inmathematics above all other great human endeavors.

The wonderful cover of this book was designed by my lifelongfriend Alexander Okun. Thank you Shurik for sharing your talentwith us. I thank David Turner and Mary Kelley, photographers of TheGazette Telegraph, for the permission to use their photographs in thisbook, and The Gazette editors for the permission to quote the articlesabout the Olympiad.

I am grateful to Philip Engel, Paul Erdos, Martin Gardner, BrankoGrünbaum, and Cecil Rousseau for kindly agreeing to be the firstreaders of the manuscript and providing me with most valuablefeedback. I am truly honored that these distinguished people havewritten introductions for this book.

I applaud my Dean James A. Null for bringing about such a climatein our College of Letters, Arts and Sciences that one is free andencouraged to create. I am grateful to my wife Maya for sustainingan intellectual atmosphere under our roof, proofreading this manu-script, and pushing me to a pen in my less inspired times. I thankSteven Bamberger, himself an award winner in the Fourth Colorado

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Mathematical Olympiad, for deciphering my manuscript andconverting it into this handsome volume.

The Colorado Mathematical Olympiad has survived for a decade.Happy Anniversary to the Olympiad and all people who made itpossible!

lii Preface

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