the ussrolympiad problem booi(

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THEUSSROLYMPIADPROBLEM OOI(SelecteclProblemsanclTheoremsof ElementaryMathematicsD.O.Shklarsky, .N.Chen zov,

and .M.YaglomThis book contains 320 unconvent ional problenrs in algebra, ari thrnct ic, clcrntntarynumber theory and trigonometry. Most of tht ' probl t ' rrrs f i rst appe:rrecl incompet i t ive examinat ions sponsorerd by thc School Matht ' rrrat i t' : rl Sot ' i t ' ty of thcMoscow State Universi ty and in the Mat l rcnrat ical ()lyrnpi l rls hckl in Most.orv.Al though most of th e problems presupp()s('orrly l rigl r st l rool rnlrl l rt ' rrrrt i t 's, t l rt 'y rrrt 'no t easy; some are of uncommon di f f icrrl ty : rrrrl wi l l cl rrlL' rrgc t l rc i rrgt 'nrri ty ol i rnyresearchmathematic ian.Nevertheless,nri r rry:rrcrvr ' l l rvi l l r i rrl l rrr' ; rt l rol rrroI i r' : rlct lhigh school students and even advanct:rl scvcrrth : rrrt l t . igl rt l r {r: r,k' rs.Th e problems ar e grouped into twclvt 's(' l ): l r: l l l scr' l i . rrs Arr, rrr11 l rtsr' ; rrr' : l l rtdivisibi l i ty of integers, equat i


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DOVER BOOKS ONMATHEMATICSA CorcrspHrsroRyoFMArrlEnaeucs, irk J. Struik. (6025b-g)$Z.gISr.lrrsrrcer-Mprnoo rnou rrrn Vmweonrror Quer,rrvCovrnol, Wtli.r,r A.Shewhart. (65232-7)$7 95Vrcrnns, Tbtsons,u.u nu B.lsrcEqumoNs oF FLUrDMEcrrANrcs.Rutherford Aris. (66110-5)98.95Tlrr TlunmrN Booxs otEucuo's Er,rrl,tnvrs,ranslated with anintroduction and commentary by Sir Thomas L. Heath.(60088-2,60089-0,60090-4)$29.85IvtnoougrroN ro P.IRTIAL F FERErtrur.Equlrroxs wrru Aler,rcertoNs;E.C. Zachmanoglouand DaIe W. Thoe. (652b1-B) 10.9bNul,mnrcer,Mrrrroos r.onScmltrlsrs .cND NcNEens, Richard Hamming.(65241-6) 15.95Onou.renyDrrrennNrrer, Equ.lrtors, Monris Tenenbaum and HarrvPollard. (64940-7)$18.95Tbowrc.lr, CercuLUSwm{ ANALyrrcGEoNc-rBy, udith L. Gersting.(67343-X)$13.95Oscrr,r,lrroxs NNorr,welln SyslEMs, ack K. Hale. (62362-6)g?.gbGpepx M.l,rrrrlarrc.c,I, Tkoucrrr ervouc OnrcrNor ArcsgRA Jacob Kle in .(27289-3)$9.95Fntrro DnrsRENcr EquerroNs,H. I-r.y & F. hssman. (67260-g)$?.gbAppuclnoNs or Fnvnr Gnor-rps,. S. Lomont. (6Z8Z6-6)gg.gsArpr,rcohoeABrr-rry Moorrs wrnr Osrur.rzltrolr Appr,rc.lrroNs.SheldonM. Ross.(67314-6)$6.95l.rlnoouctroN lr rnr Cerrulus or Vmurrons, Hans Sagan.(67366-9)$12.95IvrnoousrroN ro PenrrAr,DnrnnsNTrALEeuATroNs,Arne Broman.(66158-X)$6.95AN IrrnoousrroN tr ORDTNARyu.r.ansNTrALeuATloNS,Earl A.Coddinston. (65942-9)$8.95M.q,TRrcesND ween ThetsroRMATroNS,Charles G. Cullen.(66328-0) 8.95DrrrBnpwrnr, Fonnrswtrn Appt cATIoNS orlrE pxystcer. ScreNcEs.HarlevFlanders. (66169-5)$7.95Thnony exo Appr,rcATroNF NFrrvnpSpnres,Konrad Knopp.(66165-2) 13.95AN I*rnopuctroN nr ArcnsRArcSrRUcruREs, oseph Landin.(65940-2)$?.e5GetvmsrNoDecrsrous: IrrnooucuoN ANDCRrrrcALSr-nvpy,R. DuncanLuce and Howard Raiffa. (65943-?)$12.95Frnsr Oporn M.lrrculrrcel Locrc, Angelo Margaris. (66269-1)$?.95Ixrr.ooucrror,r ro Toeorocv, Bert Mendelson. (663b2-B) 96.9bGporvrsrRy: ColrpnprnusrveCounsn,Dan pedoe. (6b812-0)g12.gsFuNcrrowar, r.rlr,ysrs, rigyes Riesz and B6la Sz.-Nagy.(66289-6) 12.95

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TI{E USSROLYMPIADPROBLEMBOOKSelected roblcms rd Tlrcoremsof ElcmentaryMatlrcmatic

D. O. SIKLARSKYN. N. CHENTZOVI. M. YAGI.,OMREVISEDAND EDITEDBYInvnc SussMlN,Univcrsity S futrto Clara

TRANSTATEDBYJoHxMeYrovIcH,Univcrsityof funnClam

J-\ 5 fl,o".,0L.DOVERPUBLICATIONS,NC.NewYom

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PREFACE O THESECOND Russian) DITION

THp rnBssNT oLuuE,which constitutes he first part of a collection,contains 320 problems involving principally algebra and arithmetic,although severalof the problemsare of a type meantonly to encouragethe developmentof logical thought (see, or example,problems1-8).The problemsare grouped nto twelve separatesections. The lastfour sections ComplexNumbers,SomeProblems rom Number Theory,Inequalities, Numerical Sequencesnd Series) ontain important theo-retical material, and they may well serve as study topics for schoolmathematical societiesor for the Societyon ElementaryMathematicsat the pedagogicalnstitutes. In this respect he supplementary efer.encesgiven in various sectionswill also prove useful. All the othersections especiallyAlterations of Digits in Integers and Solutionsof Equations n Integers(Diophantine quations)l houldyield materialprofitable for use in mathematicsclubs and societies.Of the twelve sections, nly four (Miscellaneous roblemsn Algebra,PolynomialAlgebra, ComplexNumbers, nequalities)concernalgebra;the remaining sectionsdeal with arithmetic and number theory. Aspecialeffort has beenmade o play downproblems particularly thoseih algebra) nriolving detailid nianip-uldtivematter. This was doneto avoid dupliciting materiai in the eiCbllent ProblemBooh n Algebra.,.


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vui Olymbiad Problemsby V. A. Kretchmar (GovernmentTechnical Publishing House,Moscow1950). On the other hand, an effort has been made to render muchof the book attainable to eighth grade, and even seventh grade,students.More than three years have passed since the appearance of the firstedition of this book. During this period the original authors receiveda great many written and oral communications with respect to it,and these have been seriously considered in the reworking of thematerial and in deciding which features were worth retaining andemphasizing and which aspects were weak. As a result, the bookhas undergone considerable evision. About sixty problems that werein the first edition have been omitted-some appeared to be too diffi-cult, or were insufficiently interesting, and others did not fit intothe new structure of the book. Approximately I20 new problems havebeen added. The placing of each problem into a suitable sectionhas been restudied; the sections have been repositioned; all the solu-tions have been reworked (several were replaced by simplified orbetter solutions); and alternative solutions have been provided forsome of the problems. Hints have been given for every problem,and those problems which to the authors appear of greater difficultyhave.been starred(*). Sections 3,5,6,9, and 10 have undergone suchsignificant changes that they may be considered as having been com-pletely rewritten. Sections I,2, 4,7, and,11 have been revised radi-cally, and only Sections8 and 12 have had relatively minor alterations.The first edition of the book was prepared by I. M. Yaglom incollaboration with G. M. Adelson-Vel'sky (who contributed the sectionon alteration of digits in integers and also a number of problems toother sections, particularly to the section on Diophantine equations).An important contribution was made to the first edition by E. E.Balash (who contributed the section on numerical sequencesand series)and Y. I. Khorgin (who made the principal contribution to the sectionon inequalities). Solutions for other problems were written by variousdirectors of the School Mathematical Society of the Moscow StateUniversity. About 20 problems were taken from manuscripts of thelate D. O. Shklarsky.The rewriting of the book for the second edition was done by I.M. Yaglom, who made extensive use of the material of the first

In conclusion, the author wishes to thank A. M. Yaglom, whoseadvice was of invaluable assistance while the book was being writtenand who initiated the rewriting of the section on complex numbers.

Preface to the Second Edition ixThe author is also indebted to the editor, A. Z. Rivkin, whose nde-fatigable labors on the first and second editions made possible manyimprovements, and to all the readers who made valuable suggestions,especially I. V. Volkova, L. I. Golovina, R. S. Guter, G. Lozanovsky,I. A. Laurya, Y. B. Rutitsky, A. S. Sokolin, and I. Y. Tanatar.

I. M. Yaglom


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EDITOR'SFOREWORD O THEENGLISHEDITION

One of the important facets of science ducation n the USSR hasbeen heir series of mathematical competitive examinationsheld forstudents of high ability in the secondaryschools. Those contests,which are being emulated ncreasingly n our own educational ystem,culminate eagh year in the Soviet Union in their MathematicalOlympiads held at Moscow University, preliminary qualifying andelimination examinationshaving beenheld nationwide hroughout theacademicyear.This book, compiledover a twenty-year period, is a collection ofthe most interesting and instructive problemsposedat thesecompe-titions and in other examinationcentersof the USSR,plus additionalproblemsand material developed or use by the fthool MathematicsStudy Societies. Perhaps he greatestcomplimentwhich can be paidto the problems created or this purposeby leading Soviet mathema'ticians (or tikeii ahdadapted rom the literature) has beenthe extentto which the problems have been used in our own contests and ex'aminations.Soviet students and teachers have had available n published formthe problems,and their solutions, given in such examinations, butthis material has not generally been available in the United States.


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xrr Ofumpiad problemA few of these problems have been translated and published in suchAmerican journals as The American Mathematicat Monthly of rheMathematical Association of America, and problems of similar scopeappear as regular features of Several American journals. Except forsomecompilations from these sources, ittle exists by way of problemswhich deal with real and active mathematics instead of the fringeand recreational aspectsof the scienceor with conventional textbookexercises.This translation and revision of the Third Revisedand AugmentedEdition of the olympiad Problem Book should therefore fill a verydefinite need in American schools and colleges. It contains 320problems-a few of them merely recreational and thought-provoking,but most of them seriously engaged with solid and importantmathematical theory, albeit the preparational background is assumedto be elementary. The problems are from algebra, arithmetic,trigonometry, and number theory, and all of them emphasize thecreative aspects of these subjects. The material coordinates beauti-fully with the new conceptswhich are being emphasized n Americanschools, since the "unconventional" designation attributed to theproblems by the original authors means that they stress originalityof thought rather than mere manipulative ability and introduce thenecessity for finding new methods of attack.In this respect I am reminded of the observation made by someforgotten character in some forgotten novel who opined that theultimate test of an educative effort lay not nearly so much in whatsort of questions the students could finally answer as in what sortof questions they could finally be asked!Complete solutions to all problems are given; in many cases,alternate solutions are detailed from different points of view.Although most of the problems presuppose only high school mathe-matics, they are not in any sense easy: some are of uncommondifficulty and will challenge the ingenuity of any research mathe-matician. on the other hand, many of the problems will yield readilyto a normally bright high school student willing to use his head.Where more advanced concepts are employed, the concepts are dis-cussed in the section preceding the problems, which gives the volumethe aspect of a textbook as well as a problem book. The solutionsto more advanced problems are given in considerable detail.Hence this book can be put to use n a variety of ways for studentsof ability in high schoolsand colleges. In particular, it lends itselfexceptionally well to use in the various Institutes for high school

Editor's Foreword to the English Edition xiiimathematics teachers. It is certainly required reading for teachersdealing with the gifted student and advanced placement classes. Itwill furnish them with an invaluable fund for supplementary teachingmaterial, for self-study, and for acquiring depth in elementarymathematics.Except for the elimination of the few misprints and errors foundin the original, and some recasting of a few proofs which did notappear to jell when translated literally, the translation is a faithfulone: it was felt that the volume would lose something by too muchtampering. (For this reason the original foreword and preface havealso been retained). Thus the temptation to radically alter or simplifyany understandable solution was resisted (as, for example, in thesections on number theory and inequalities, where congruencearithmetic would certainly have supplied some neater and more directproofs). Some notations which differ in minor respects from thestandard American notations have been retained (as, for example,C* instead of Cl). These will cause no difficulty.All references made in the text to books not available in Englishtranslation have been retained; no one can know when translationsof some of those volumes will appear. Whenever an English trans'lation was known to exist, the translated edition is referred to.The translation was made from the Third (Russian) Edition ofSelectedProblems and Theorems of Elernentary Mathernatics, whichis the title under which the original volume appeared in the SovietUnion. Mr. John Maykovich, instructor at the University of SantaClara, was the translator, and he was assisted by Mrs. Alvin (Myra)White, who translated fifty pages. The writing out, revising, editing,annotating, and checking against the original Russian were by myown hand.Thanks are due the following persons or their assistance n readingportions of the translation, pointing out errors, and making valuablesuggestions: Professor George Polya of Stanford University,r ProfessorAbraham Hillman of the University of Santa Clara, and ProfessorRobert Rosenbaum of Wesleyan University.I shall be very grateful to readers who are kind enough to pointout errors, misprints, misleading statements of problems, and in-correct or obscure proofs found in this edition.January 1962 Irving Sussman

t I would also like to call attention to Professor Polya's new book Mathtmotinal Di,saaerg (Wiley) which contains elementary problems and valuabletextual discussion of approaches to, and techniques of, problem solving'


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&

CONTENTS

Foreword to the Third (Russian)EditionPreface to the Second(Russian)EditionEditor's Foreword to the English EtlitionFrom the AuthorsSuggestions or Using this BookNumerical Reference o the ProblemsGiven in the Moscow

Mathematical Olympiads1. Introductory Problems(1-14)2. Alterations of Digits in Integers (15-26)3. The Divisibility of Integers (27-7I)4. SomeProblems rom Arithmetic (72-109)5. Equations Having Integer Solutions (110-130)6. Evaluating Sumsand Products (131-159)7. MiscellaneousProblems from Algebra (1@-195)8. The Algebra of Polynomials(19G22f)9. Complex Numbers (Zn-29)

vvi ixi

I3c6111320273038,t5

50xv


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xvi10.11.t2 .

Some Problems of Number Theory en-ZS4)SomeDistinctive Inequalities (2S$-30g)Difference Sequences nd Sums (309-320)

Ollttnfiad Problems56617480423

SolutionsAnswers and Hints

FROM THE AUTHORS

THsrHnprvoLUuEsthatmakeupthepresentcollect ionofproblemsare the commencementof a series of books based on materialg"trr"i"a by the school Mathematics society of.the MoscowstateUniversityoveratwenty.yearperiod.Thetextconsistsofproblemsand theorems, most of which have been presentgdduring meetingsof the various sectionsof the School Mathematical Society of theIA.S.U. as well as in the Mathematical Olympiads held in Moscow'(The numbersof the problemsgiven in the Olympiadsare listed onp. 5).These volumesare directed to students, teachers,and directorsofschoolmathematical ocieties nd societies n elementarymathematicsoi tf," pedagogical nstitutes. The first volume (Part I) containsproblems n arittrmetic, algebra, and number theory' The secondvolume is devoted o problems n plane geometry, and the third toproblems n solid geometry.Incontrast tothemajor i tyofproblembooksintendedforhighschoolstudents, thesebooksare designednot only to reinforce thestudent's formal knowledge,but also to acquaint him with methodsand deasnew to him and to develophis predilection for, and abilityin, original thinking. Here, there are few problemswhosesolutionsI


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the ussrolympiad problem booi(

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