In this post, we will see the two volume set of Challenging Mathematical Problems With Elementary Solution by A. M. Yaglom; I. M. Yaglom .

Volume 1: Combinatorial Analysis and Probability Theory

Volume 2: Problems From Various Branches of Mathematics

# About the books

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 excellent 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: combinatorial 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.

Designed for advanced high school students, undergraduates, graduate students, mathematics teachers and any lover of mathematical challenges, this two-volume set offers a broad spectrum of challenging problems—ranging from relatively simple to extremely difficult. Indeed, some rank among the finest achievements of outstanding mathematicians.

Translated from a well-known Russian work entitled Non-Elementary Problems in an Elementary Exposition, the chief aim of the book is to acquaint the reader with a variety of new mathematical facts, ideas and methods. And while the majority of the problems represent questions in higher (“non-elementary”) mathematics, most can be solved with elementary mathematics. In fact, for the most part, no knowledge of mathematics beyond a good high school course is required.

Volume Two contains 74 problems from various branches of mathematics, dealing with such topics as points and lines, lattices of points in the plane, topology, convex polygons, distribution of objects, non-decimal counting, theory of primes and more. In both volumes 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

Ideal as a textbook, for self-study, or as a working resource for a mathematics club, this wide-ranging compilation offers 174 carefully chosen problems that will test the mathematical acuity and problem-solving skills of almost any student, teacher or mathematician.

Volume 1 was translated from Russian by James McCawley Jr. was published in 1964.

Volume 2 was translated from Russian by James McCawley Jr. was published 1967.

Credits to original uploader.

You can get Volume 1 here.

You can get Volume 2 here.

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Add new entries to the detailed book catalog here.

# Contents

# Volume 1

## PROBLEMS

I. Introductory problems 4

II. 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 15

VI. Problems on computing probabilities 20

VII. Experiments with infinitely many possible outcomes 27

VIII. Experiments with a continuum of possible outcomes 30

## SOLUTIONS

I. Introductory problems 39

II. The representation of integers as sums and products 52

III. Combinatorial problems on the chessboard 76

IV. Geometric problems on combinatorial analysis 102

V. Problems on the binomial coefficients 125

VI. Problems on computing probabilities 141

VII. Experiments with infinitely many possible outcomes 194

VIII. Experiments with a continuum of possible outcomes 211

# Volume 2

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 11

V. Convex Polygons 11

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 𝜋 22

XI. The Calculation of Areas of Regions Bounded by Curves 33

XII. Some Remarkable Limits 38

XIII. The Theory of Primes 45

Solutions 45

Hints and Answers 199

Bibliography 213

If you want a hard copy version of these and other books by Yaglom, you can get a reprint from Dover Books in the US:

https://doverpublications.ecomm-search.com/m?formSubmitted=true&keywords=Yaglom&x=0&y=0

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