Table Of ContentA. 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
