THE USSROLYMPIADPROBLEM BOOI(

Selectecl Problems ancl Theoremsof Elementary Mathematics

D.O. Shklarsky, N. N. Chen tzov,and l.M.Yaglom

This book contains 320 unconventional problenrs in algebra, arithrnctic, clcrntntarynumber theory and trigonometry. Most of tht' problt 'rrrs first appe:rrecl in

competit ive examinations sponsorerd by thc School Matht'rrratit ' :r l Sot' it ' ty of thcMoscow State Universi ty and in the Mat l rcnrat ical ( ) lyrnpi l r ls hckl in Most.orv.

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OLYMPIAD

Selected Problems

THE USSR

and Theorems of

PROBLEM

ElementaryMathematiCs

BOOI(

D.O. Shklarsl<y, N.N. Chentzovand l.M.Yaglom

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(antinu.edon fuck flnp)

TI{E USSR OLYMPIADPROBLEM BOOKSelected Problcms ard Tlrcorems

of Elcmentary Matlrcmatic s

D. O. SIKLARSKYN. N. CHENTZOV

I. M. YAGI.,OMREVISED AND EDITED BY

Invnc SussMlN,Univcrsity S futrto Clara

TRANSTATED BYJoHx MeYrovIcH,

Univcrsity of funnClam

J-\ 5 fl, o".,0 iL. , '

DOVER PUBLICATIONS, INC.New Yom

v-L ! "/ oo

Published in Canada by General hrblishing Company, Ltd., 30 Lesmill Road,Don Mills, Toronto, Ontario.

hrblished in the United Kingdom by Constable and Company, Ltd., 3 TheLanchesters, 162-l& ftrlham Falace Road, London W6 9ER.

Bibliographical Note

This Dover edition, first published in 1993, is an unabridged and unalteredrepublication of the work finst publish€d by W.H. freeman and Company, Sanfrancisco, in1962.

Library of Congress Cataloging-in-Publication Dan

Shkliarskil, D. O. (David Oskarwich), lgt&-1942.fizbrannp zadrchi i tcoremy elementarnol matematiki, ch l. English]The USSR Olympiad problem book : selecrcd problems and-theorems of

elementary mathematics / D.O. Shklarsky, N.N. Chentzor,, I.M. yaglom; trans-lated by John Maykorrich.-3rd ed / rcv. and edited by Irving Sussman.

p. cm.rsBN G48G27709-7l. Mathematics-hoblems, exercises, etc. I. Chentsov, N. N. (Nikolal

Nikolaevich) tr. IAglom, I. M. (lsaak Moiseevich), l92l-. III. Sussman,

FOREWORD TO THE

THIRD (Russian) EDITION

Tnrs aoox coNrAINs 320 unconventional problems in algebra, arithme'tic, elementary number theory, and trigonometry. Most of theseproblems first appeared in competitive examinations sponsored by theSchool Mathematical Society of the Moscow State University and inthe Mathematical Olympiads held in Moscow. The book is designedfor students having a mathematical background at the high sghogllevel;r very many of the problems are within reach of seventh&6ndeighth grhde students of outstanding ability. Solutions are given

for all the problems. The solutions for the more difficult problems

are especially detailed.The third (Russian) edition differs from the second chiefly in the

elimination of errors detected in the second edition. Therefore, thepreface to the second edition is retained.

t The level of academic attainment referred to as "high school level" is the

American ninth to twelfth grades. The USSR equivalent is seventh to tenthgrades. This means that this material is introduced about two years earlierin the Russian schools. Since Russian children begin their first grade studiesabout a year later than do American children, the actual age disparity is not

as much as two years l0ditorl.v

i l

Irving. IV. Trtle.QA43.S5813 19945lO'.7*dc20 93-11553

CIP

Manufactured in the United States of AmericaDover Publications, Inc., 3l East 2nd Street, Mineola N.y. ll50l

PREFACE TO THE

SECOND (Russian) EDITION

THp rnBssNT voLuuE, which constitutes the first part of a collection,contains 320 problems involving principally algebra and arithmetic,although several of the problems are of a type meant only to encouragethe development of logical thought (see, for example, problems 1-8).

The problems are grouped into twelve separate sections. The lastfour sections (Complex Numbers, Some Problems from Number Theory,Inequalities, Numerical Sequences and Series) contain important theo-retical material, and they may well serve as study topics for schoolmathematical societies or for the Society on Elementary Mathematicsat the pedagogical institutes. In this respect the supplementary refer.ences given in various sections will also prove useful. All the othersections [especially Alterations of Digits in Integers and Solutionsof Equations in Integers (Diophantine equations)l should yield materialprofitable for use in mathematics clubs and societies.

Of the twelve sections, only four (Miscellaneous Problems in Algebra,Polynomial Algebra, Complex Numbers, Inequalities) concern algebra;the remaining sections deal with arithmetic and number theory. Aspecial effort has been made to play down problems (particularly thoseih algebra) inriolving detailid nianip-uldtive matter. This was doneto avoid dupliciting materiai in the eiCbllent Problem Booh in Algebra.,.

vui Olymbiad Problems

by V. A. Kretchmar (Government Technical 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 revision. 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 Sections 8 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 sequences and 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 ix

The author is also indebted to the editor, A. Z. Rivkin, whose inde-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

EDITOR'S FOREWORD TO THE

ENGLISH EDITION

One of the important facets of science education in the USSR hasbeen their series of mathematical competitive examinations held forstudents of high ability in the secondary schools. Those contests,which are being emulated increasingly in our own educational system,culminate eagh year in the Soviet Union in their MathematicalOlympiads held at Moscow University, preliminary qualifying andelimination examinations having been held nationwide throughout theacademic year.

This book, compiled over a twenty-year period, is a collection ofthe most interesting and instructive problems posed at these compe-titions and in other examination centers of the USSR, plus additionalproblems and material developed for use by the fthool MathematicsStudy Societies. Perhaps the greatest compliment which can be paidto the problems created for this purpose by leading Soviet mathema'ticians (or tikeii ahd adapted from the literature) has been the extentto which the problems have been used in our own contests and ex'aminations.

Soviet students and teachers have had available in published formthe problems, and their solutions, given in such examinations, butthis material has not generally been available in the United States.

xrr Ofumpiad problem

A 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 forsome compilations from these sources, little exists by way of problemswhich deal with real and active mathematics instead of the fringeand recreational aspects of the science or with conventional textbookexercises.

This translation and revision of the Third Revised and 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 concepts which are being emphasized in 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 in a variety of ways for studentsof ability in high schools and colleges. In particular, it lends itselfexceptionally well to use in the various Institutes for high school

Editor's Foreword to the English Edition xiii

mathematics teachers. It is certainly required reading for teachers

dealing with the gifted student and advanced placement classes. It

will furnish them with an invaluable fund for supplementary teaching

material, for self-study, and for acquiring depth in elementarymathematics.

Except for the elimination of the few misprints and errors found

in the original, and some recasting of a few proofs which did not

appear to jell when translated literally, the translation is a faithful

one: it was felt that the volume would lose something by too much

tampering. (For this reason the original foreword and preface have

also been retained). Thus the temptation to radically alter or simplify

any understandable solution was resisted (as, for example, in the

sections on number theory and inequalities, where congruence

arithmetic would certainly have supplied some neater and more directproofs). Some notations which differ in minor respects from the

standard 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 English

translation have been retained; no one can know when translations

of 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 of

Selected Problems and Theorems of Elernentary Mathernatics, which

is the title under which the original volume appeared in the Soviet

Union. Mr. John Maykovich, instructor at the University of Santa

Clara, 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 my

own hand.Thanks are due the following persons for their assistance in reading

portions of the translation, pointing out errors, and making valuable

suggestions: Professor George Polya of Stanford University,r Professor

Abraham Hillman of the University of Santa Clara, and Professor

Robert Rosenbaum of Wesleyan University.I shall be very grateful to readers who are kind enough to point

out errors, misprints, misleading statements of problems, and in-

correct or obscure proofs found in this edition.

January 1962 Irving Sussmant I would also like to call attention to Professor Polya's new book Matht

motinal Di,saaerg (Wiley) which contains elementary problems and valuabletextual discussion of approaches to, and techniques of, problem solving'

&

CONTENTS

Foreword to the Third (Russian) EditionPreface to the Second (Russian) EditionEditor's Foreword to the English EtlitionFrom the AuthorsSuggestions for Using this BookNumerical Reference to the Problems Given 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. Some Problems from Arithmetic (72-109)

5. Equations Having Integer Solutions (110-130)

6. Evaluating Sums and Products (131-159)7. Miscellaneous Problems from Algebra (1@-195)

8. The Algebra of Polynomials (19G22f)

9. Complex Numbers (Zn-29)

vviixiI3

c

6111320273038,t550xv

xvi

10.11.t2.

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

Ollttnfiad Problems

56617480

423

SolutionsAnswers and Hints

FROM THE AUTHORS

THsrHnprvoLUuEsthatmakeupthepresentcol lect ionofproblemsare the commencement of a series of books based on material

g"trr"i"a by the school Mathematics society of.the Moscow state

Universi tyoveratwenty.yearperiod.Thetextconsistsofproblemsand theorems, most of which have been presentgd during meetings

of the various sections of the School Mathematical Society of the

IA.S.U. as well as in the Mathematical Olympiads held in Moscow'

(The numbers of the problems given in the Olympiads are listed on

p. 5).These volumes are directed to students, teachers, and directors of

school mathematical societies and societies on elementary mathematics

oi tf," pedagogical institutes. The first volume (Part I) contains

problems in arittrmetic, algebra, and number theory' The second

volume is devoted to problems in plane geometry, and the third to

problems in solid geometry.Incontrast tothemajor i tyofproblembooksintendedforhigh

school students, these books are designed not only to reinforce the

student's formal knowledge, but also to acquaint him with methods

and ideas new to him and to develop his predilection for, and ability

in, original thinking. Here, there are few problems whose solutionsI