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The Colorado Mathematical Olympiadand Further Explorations

Alexander SoiferThe Colorado MathematicalOlympiad and FurtherExplorationsFrom the Mountains of Coloradoto the Peaks of MathematicsWith over 185 IllustrationsForewords byPhilip L. EngelPaul ErdősMartin GardnerBranko GrünbaumPeter D. Johnson, Jr.and Cecil Rousseau123

Alexander SoiferCollege of Letters, Arts and SciencesUniversity of Colorado at Colorado Springs1420 Austin Bluffs ParkwayColorado Springs, CO 80918USAasoifer@uccs.eduISBN: 978-0-387-75471-0e-ISBN: 978-0-387-75472-7DOI: 10.1007/978-0-387-75472-7Springer New York Dordrecht Heidelberg LondonLibrary of Congress Control Number: 2011925380c Alexander Soifer 2011 All rights reserved. This work may not be translated or copied in whole or in part without the writtenpermission of the publisher (Springer Science Business Media, LLC, 233 Spring Street, New York,NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Usein connection with any form of information storage and retrieval, electronic adaptation, computersoftware, or by similar or dissimilar methodology now known or hereafter developed is forbidden.The use in this publication of trade names, trademarks, service marks, and similar terms, even if theyare not identified as such, is not to be taken as an expression of opinion as to whether or not they aresubject to proprietary rights.Printed on acid-free paperSpringer is part of Springer Science Business Media (www.springer.com)

To all those peoplethroughout the world,who create Olympiadsfor new generations of mathematicians.

Forewords to The Colorado MathematicalOlympiad and Further ExplorationsWe live in an age of extreme specialization – in mathematicsas well as in all other sciences, in engineering, in medicine.Hence, to say that probably 90% of mathematicians cannotunderstand 90% of mathematics currently published is, mostlikely, too optimistic. In contrast, even a pessimist would haveto agree that at least 90% of the material in this book is readily accessible to, and understandable by 90% of students inmiddle and high schools. However, this does not mean thatthe topics are trivial – they are elementary in the sense thatthey do not require knowledge of lots of previously studiedmaterial, but are sophisticated in requiring attention, concentration, and thinking that is not fettered by preconceptions.The organization in groups of five problems for each of the“Olympiads”, for which the participants were allowed fourhours, hints at the difficulty of finding complete solutions.I am convinced that most professional mathematicians wouldbe hard pressed to solve a set of five problems in two hours,or even four.There are many collections of problems, for “Olympiads”of various levels, as well as problems in a variety of journals.What sets this book apart from the “competition” are several aspects that deserve to be noted.vi

Forewordsvii The serenity and enthusiasm with which the problems, andtheir solutions, are presented; The absence of prerequisites for understanding the problems and their solutions; The mixture of geometric and combinatorial ideas that arerequired in almost all cases.The detailed exposition of the trials and tribulations endured by the author, as well as the support he received, shedlight on the variety of influences which the administration ofa university exerts on the faculty. As some of the negative actions are very probably a consequence of mathophobia, thespirit of this book may cure at least a few present or futuredeciders from that affliction.Many mathematicians are certainly able to come up withan interesting elementary problem. But Soifer may be uniquein his persistence, over the decades, of inventing worthwhileproblems, and providing amusing historical and other comments, all accessible to the intended pre-college students.It is my fervent hope that this book will find the widereadership it deserves, and that its readers will feel motivatedto look for enjoyment in mathematics.Branko GrünbaumDepartmernt of MathematicsUniversity of Washington, Seattle

Here is another wonderful book from Alexander Soifer. Thisone is a more-than-doubling of an earlier book on the first10 years of the Colorado Mathematical Olympiad, whichwas founded and nourished to robust young adulthood by–Alexander Soifer.Like The Mathematical Coloring Book, this book is not somuch mathematical literature as it is literature built aroundmathematics, if you will permit the distinction. Yes, there isplenty of mathematics here, and of the most delicious kind.In case you were unaware of, or had forgotten (as I had), thelevel of skill, nay, art, necessary to pose good olympiad- orPutnam exam-style problems, or the effect that such a problem can have on a young mind, and even on the thoughts ofa jaded sophisticate, then what you have been missing can befound here in plenty – at least a year’s supply of great intellectual gustation. If you are a mathematics educator looking foractivities for a math. club – your search is over! And with theFurther Explorations sections, anyone so inclined could spenda lifetime on the mathematics sprouting from this volume.But since there will be no shortage of praise for the mathematical and pedagogical contributions of From the Mountainsof Colorado. . . , let me leave that aspect of the work and supplyviii

Forewordsixa few words about the historical account that surrounds andbinds the mathematical trove, and makes a story of it all. TheHistorical Notes read like a war diary, or an explorer’s letters home: there is a pleasant, mundane rhythm of reportage– who and how many showed up from where, who the sponsors were, which luminaries visited, who won, what theirprizes were – punctuated by turbulence, events ranging fromA. Soifer’s scolding of a local newspaper for printing onlythe names of the top winners, to the difficulties arising fromthe weather and the shootings at Columbine High School in1999 (both matters of life and death in Colorado), to the inexplicable attempts of university administrators to impede,restructure, banish, or destroy the Colorado MathematicalOlympiad, in 1985, 1986, 2001 and 2003. It is fascinating stuff.The very few who have the entrepeneurial spirit to attemptthe creation of anything like an Olympiad will be forewarnedand inspired.The rest of us will be pleasurably horrified and amazed,our sympathies stimulated and our support aroused for thebrave ones who bring new life to the communication ofmathematics.Peter D. Johnson. Jr.Department of Mathematics and StatisticsAuburn University

In the common understanding of things, mathematics is dispassionate. This unfortunate notion is reinforced by modernmathematical prose, which gets good marks for logic andpoor ones for engagement. But the mystery and excitementof mathematical discovery cannot be denied. These qualitiesoverflow all preset boundaries.On July 10, 1796, Gauss wrote in his diaryEΥPHKA! num Δ Δ ΔHe had discovered a proof that every positive integer is thesum of three triangular numbers {0, 1, 3, 6, 10, . . . n(n 1)/2, . . .}. This result was something special. It was right to celebrate the moment with an exclamation of Eureka!In 1926, an intriguing conjecture was making the roundsof European universities.If the set of positive integers is partitioned into twoclasses, then at least one of the classes contains an nterm arithmetic progression, no matter how large nis taken to be.x

ForewordsxiThe conjecture had been formulated by the Dutchmathematician P. J. H. Baudet, who told it to his friend andmentor Frederik Schuh.B. L. van der Waerden learned the problem in Schuh’sseminar at the University of Amsterdam. While in Hamburg,van der Waerden told the conjecture to Emil Artin and OttoSchreier as the three had lunch. After lunch, they adjournedto Artin’s office at the University of Hamburg to try to finda proof. They were successful, and the result, now knownas van der Waerden’s Theorem, is one of the Three Pearls ofNumber Theory in Khinchine’s book by that name. The storydoes not end there. In 1971, van der Waerden published a remarkable paper entitled How the Proof of Baudet’s Conjecturewas Found.1 In it, he describes how the three mathematicianssearched for a proof by drawing diagrams on the blackboardto represent the classes, and how each mathematician hadEinfälle (sudden ideas) that were crucial to the proof. In thisaccount, the reader is a fourth person in Artin’s office, observing with each Einfall the rising anticipation that the proof isgoing to work. Even though unspoken, each of the three musthave had a “Eureka moment” when success was assured.From the Mountains of Colorado to the Peaks of Mathematics presents the 20-year history of the Colorado Mathematical Olympiad. It is symbolic that this Olympiad is held inColorado. Colorado is known for its beauty and spaciousness.In the book there is plenty of space for mathematics. There arewonderful problems with ingenious solutions, taken from geometry, combinatorics, number theory, and other areas. Butthere is much more. There is space to meet the participants,hear their candid comments, learn of their talents, mathematical and otherwise, and in some cases to follow their pathsas professionals. There is space for poetry and references tothe arts. There is space for a full story of the competition – its1Studies in Pure Mathematics (Presented to Richard Rado), L. Mirsky, ed., Academic Press, London, 1971, pp 251-260.

xiiForewordsdreams and rewards, hard work and conflict. There is spacefor the author to comment on matters of general concern. Onesuch comment expresses regret at the limitations of currentlyaccepted mathematical prose.In my historical-mathematical research for TheMathematical Coloring Book, I read a good numberof nineteenth-century Victorian mathematical papers. Clearly, the precision and rigor of mathematical prose has improved since then, but somethingcharming was lost – perhaps, we lost the “tasteof time” in our demand for an “objective,” impersonal writing, enforced by journal editors and manypublishers. I decided to give a historical taste tomy Olympians, and show them that behind Victorian clothing we can find the pumping heart of theOlympiad spirit. [p. 235]Like Gauss, Alexander Soifer would not hesitate to injectEureka! at the right moment. Like van der Waerden, he cantransform a dispassionate exercise in logic into a compellingaccount of sudden insights and ultimate triumph.Cecil RousseauProfessor EmeritusUniversity of MemphisChair, USA Mathematical OlympiadCommittee

Forewords to Colorado MathematicalOlympiadLove! Passion! Intrigue! Suspense! Who would believe thatthe history of a mathematics competition could accurately bedescribed by words that more typically appear on the back ofa popular novel? After all, mathematics is dull; history is dull;school is dull. Isn’t that the conventional wisdom?In describing the history of the Colorado MathematicalOlympiad, Alexander Soifer records the comments of a mathematics teacher who anonymously supported the Olympiadin each of its first ten years. When asked why, this unselfishteacher responded “I love my profession. This is my way togive something back to it.” Alexander also loves his profession. He is passionate about his profession. And he workshard to give something back.The Colorado Mathematical Olympiad is just one wayAlexander demonstrates his love for mathematics, his love forteaching, his love for passing on the incredible joy of discovery. And as you read the history of the Olympiad, you cannothelp but be taken up yourself with his passion.But where there is passion, there is frequently intrigue.Here it involves the efforts of school administrators and others to help – or to hinder – the success of the Olympiad. ButAlexander acknowledges that we all must have many friendsxiii