Challenging mathematical problems with elementary solutions [Vol. II] PDF
Preview Challenging mathematical problems with elementary solutions [Vol. II]
A. M. Yaglom and l. M. Yaglom CHALLENGING MATHEMATICAL PROBLEMS WITH ELE:MENTARY SOLlmONS Volume II Problems From Various Branches of Mathematics Translated by James McCawley, Jr. Revised and edited by Basil Gordon DOVER PUBLICATIONS, INC. NEW YORK Copyright © 1967 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 1967. It was published then as part of the Survey of Recent East European Mathematical Literature. a project conducted by Alfred L. Putnam and Izaak Wirs7Up. 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 Neelementarnre Zadac-hi 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 Yaglom and 1M. Yaglom : 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. I 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. QAI64.11613 1987 511'.6-dcI9 87-27298 ClP PREFACE TO THE AMERICAN EDITION As in the case of Volume I, a considerable number of changes have been made in adapting the original book for use by English-speaking readers. The editor accepts full responsibility for these changes which include the following: 1. The order of a few problems has been changed in order that no problem need depend on a later one for its solution. 2. Problems 110 and 128 were added in order to supplement and lend perspective to problems 112 and 127, respectively. 3. Several solutions, and some of the introductory paragraphs, have been expanded or rewritten to bring out points not familiar to many American readers. This applies in particular to most of Section 8. 4. The bibliography has been considerably enlarged. BASIL GORDON Cambridge 1967 v SUGGESTIONS FOR USING THE BOOK This volume contains seventy-four 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 125 and 170, and, in general, all problems marked with three asterisks. 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. In this case the additional literature cited in the text will be of value. While the easier problems could be solved by the partic ipants 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. CONTENTS Preface to the American Edition v Suggestions for Using the Book vii Problems 3 I. Points and Lines 3 II. Lattices of Points in the Plane 5 III. Topology 7 IV. A Property of the Reciprocals of Integers II V. Convex Polygons II VI. Some Properties of Sequences of Integers 12 VII. Distribution of Objects 13 VIII. Nondecimal Counting 13 IX. Polynomials with Minimum Deviation from Zero (Tchebychev Polynomials) 20 X. Four Formulas for 22 11" XI. The Calculation of Areas of Regions Bounded by Curves 25 XII. Some Remarkable Limits 33 XIII. The Theory of Primes 38 Solutions 45 Hints and Answers 199 Bibliography 213 ix CHALLENGING MATHEMATICAL PROBLEMS WITH ELEMENTARY SOLlJrIONS Volume II Problems From Various Branches of Mathematics
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