Challenging mathematical problems with elementary solutions [Vol. I] PDF
Preview Challenging mathematical problems with elementary solutions [Vol. I]
A. M. Yag!om and l. M. Yag!om CHALLENGING MATHEMATICAL PROBLEMS WITH ELEMENTARY SOLlJrIONS Volume I Combinatorial Analysis and Probability Theory Translated by James McCawley, Jr. Revised and edited by Basil Gordon DOVER PUBLICATIONS, INC. NEW YORK Copyright © 1964 by The University of Chicago. All rights reserved under Pan American and International Copyright Conventions. Published in Canada by General Publishing Company, Ltd., 30 Lesmill Road, Don Mills, Toronto, Ontario. Published in the United Kingdom by Constable and Com pany, Ltd., 10 Orange Street, London WC2H 7EG. This Dover edition, first published in 1987, is an un abridged and unaltered republication of the edition pub lished by Holden-Day, Inc., San Francisco, in 1964. It was published then as part of the Survey of Recent East European Mathematical Literature. a project conducted by Alfred L. Putnam and Izaak Wirszup, Dept. of Mathematics, The University of Chicago, under a grant from The National Science Foundation. It is reprinted by special arrangement with Holden-Day, Inc., 4432 Telegraph Ave., Oakland, California 94609. Originally published as Neelementarnye Zadachi v Ele mentarnom Izlozhenii by the Government Printing House for Technical-Theoretical Literature, Moscow, 1954. Manufactured in the United States of America Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501 Library of Congress Cataloging-in-Publication Data Yaglom, A. M. [Neelementarnye zadachi v elementarnom izlozhenii. English] Challenging mathematical problems with elementary solutions I A. M. YagJom and I. M. YagJom: translated by James McCawley, Jr. : revised and edited by Basil Gordon. p. cm. Translation of: Neelementarnye zadachi v elementar nom izlozhenii. Reprint. Originally: San Francisco : Holden-Day, 1964-1967. Bibliography: p. Includes indexes. Contents: v. 1. Combinatorial analysis and probability theory-v. 2. Problems from various branches of math ematics. ISBN 0-486-65536-9 (pbk. : v. I). ISBN 0-486-65537-7 (pbk. : v. 2) I. Combinatorial analysis-Problems, exercises, etc. 2. Probabilities-Problems, exercises, etc. 3. Math- ematics-Problems, exercises, etc. I. Yaglom, I. M. (Isaac Moiseevich), 1921- II. Gordon, Basil. III. Title. QA 164.11613 1987 511'.6-dcI9 87-27298 CIP PREFACE TO THE AMERICAN EDITION This book is the first of a two-volume translation and adaptation of a well-known Russian problem book entitled Non-Elementary Problems in an Elementary Exposition. * The first part of the original, Problems on Combinatorial Analysis and Probability Theory, appears as Volume I, and the second part, Problems from Various Branches of Mathematics, as Volume II. The authors, Akiva and Isaak Yaglom, are twin brothers, prominent both as mathematicians and as expositors, whose many excel lent books have been exercising considerable influence on mathematics education in the Soviet Union. This adaptation is designed for mathematics enthusiasts in the upper grades of high school and the early years of college, for mathematics instructors or teachers and for students in teachers' colleges, and for all lovers of the discipline; it can also be used in problem seminars and mathematics clubs. Some of the problems in the book were originally discussed in sections of the School Mathematics Circle (for secondary school students) at Moscow State University; others were given at Moscow Mathematical Olympiads, the mass problem-solving contests held annually for mathematically gifted secondary school students. The chief aim of the book is to acquaint the reader with a variety of new mathematical facts, ideas, and methods. The form of a problem book has been chosen to stimulate active, creative work on the materials presented. The first volume contains 100 problems and detailed solutions to them. Although the problems differ greatly in formulation and method of solution, they all deal with a single branch of mathematics: combina torial analysis. While little or no work on this subject is done in American high schools, no knowledge of mathematics beyond what is imparted in a good high school course is required for this book. The authors have tried to outline the elementary methods of combinatorial analysis with some completeness, however. Occasionally, when needed, additional explanation is given before the statement of a problem. • Neelementarnye zadachi " eiementarnom izlozhenii, Moscow: Gostekhizdat, 19S4. II vi Preface Thus the majority of the problems in this book and in its companion volume represent questions in higher (Unon-elementary") mathematics that can be solved with elementary mathematics. Most of the problems in this volume are not too difficult and resemble problems encountered in high school. The last three sections, however, contain some very difficult problems. Before going on to the problems, the reader should consult the uSuggestions for Using the Book." The book was translated by Professor James McCawley, Jr., of the University of Chicago and edited and revised by Professor Basil Gordon of the University of California at Los Angeles. Problem 85 was sent by the Russian authors for inclusion in the American edition, and appears here for the first time. A number of revisions have been made by the editor: I. In order to make volume I self-contained, some problems were transferred to volume II. To replace these, problems 1,3,12, and 100 were added. Problem 12, in which the principle of inclusion and exclusion is presented, is intended to unify the treatment of several subsequent problems. 2. Some of the problems have been restated in order to illustrate the same ideas with smaller numbers. 3. The introductory remarks to section I, 2, 6, and 8 have been rewritten so as to explain certain points with which American readers might not be familiar. 4. Adaptation of this book for American use has involved these customary changes: References to Russian money, sports, and so forth have been converted to their American equivalents; some changes in notation have been made, such as the introduction of the notation of set theory where appropriate; some comments dealing with personalities have been deleted; and Russian biblio graphical references have been replaced by references to books in English, whenever possible. The editor wishes to thank Professor E. G. Straus for his helpful suggestions made during the revision of the book. The Survey wishes to express its particular gratitude to Professor Gordon for the valuable improvements he has introduced. SUGGESTIONS FOR USING THE BOOK This book contains one hundred problems. The statements of the problems are given first, followed by a section giving complete solutions. Answers and hints are given at the end of the book. For most of the problems the reader is advised to find a solution by himself. After solving the problem, he should check his answer against the one given in the book. If the answers do not coincide, he should try to find his error; if they do, he should compare his solution with the one given in the solutions section. If he does not succeed in solving the problem alone, he should consult the hints in the back of the book (or the answer, which may also help him to arrive at a correct solution). If this is still no help, he should turn to the solution. It should be emphasized that an attempt at solving the problem is of great value even if it is unsuccessful: it helps the reader to penetrate to the essence of the problem and its difficulties, and thus to understand and to appreciate better the solution presented in the book. But this is not the best way to proceed in all cases. The book con tains many difficult problems, which are marked, according to their difficulty, by one, two, or three asterisks. Problems marked with two or three asterisks are often noteworthy achievements of outstanding mathe maticians, and the reader can scarcely be expected to find their solutions entirely on his own. It is advisable, therefore, to turn straight to the hints in the case of the harder problems; even with their help a solution will, as a rule, present considerable difficulties. The book can be regarded not only as a problem book, but also as a collection of mathematical propositions, on the whole more complex than those assembled in Hugo Steinhaus's excellent book, Mathematical Snapshots (New York: Oxford University Press, 1960), and presented in the form of problems together with detailed solutions. If the book is used in this way, the solution to a problem may be read after its statement is clearly understood. Some parts of the book, in fact, are so written that this is the best way to approach them. Such, for example, are problems 53 and 54, problems 83 and 84, and, in general, all problems marked with three asterisks. Sections VII and VIII could also be treated in this way. vii viii Suggestions for using the book The problems are most naturally solved in the order in which they occur. But the reader can safely omit a section he does not find interesting. There is, of course, no need to solve all the problems in one section before passing to the next. This book can well be used as a text for a school or undergraduate mathematics club studying combinatorial analysis and its applications to probability theory. In this case the additional literature cited in the text will be of value. While the easier problems could be solved by the partie pants alone, the harder ones should be regarded as "theory." Their solutions might be studied from the book and expounded at the meetings of the club. INDEX OF PROBLEMS GIVEN IN THE MOSCOW MATHEMATICAL OLYMPIADS The Olympiads are conducted in two rounds: the first is an elimination round, and the second is the core of the competition. I I Olympiads Round I Round II Olympiads Round I Round 11 For 7th and 8th graders For 9th and 10th graders VI (1940) - 16, 35a 1(1935) - 6,27 VIII (1945) - 62a II (1936) - 17 X (1947) 20 - III (1937) - 47 XIII (1950) - 541 IV (1938) 2 13a,45a V (1939) - 451Y VI (1940) 4 15 VIII (1945) - 62b X (1947) 49a - Xl (1948) - 26 XII (1949) - 91a 1 For n = 10. 2 For n = 5. PROBLEMS J. rntroductory problems - 4 I r. The representation of integers as sums and products - 5 III. Combinatorial problems on the chessboard - 10 IV. Geometric problems on combinatorial analysis - 12 V. Problems on the binomial coefficients - ] 5 VI. Problems on computing probabilities - 20 VB. Experiments with infinitely many possible outcomes - 27 . VnL Experiments with a continuum of possible outcomes - 30 CHALLENGING MATHEMATICAL PROBLEMS WITH ELEMENTARY SOLurIONS Volume I Combinatorial Analysis and Probability Theory
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