Table Of ContentA. 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
