Challenging Problems in Algebra

ALFRED S. POSAMENTIER Professor of Mathematics Education

The City College of the City University of New York

CHARLES T. SALKIND Late Professor of Mathematics

Polytechnic University, New York

DOVER PUBLICATIONS, INC. New York

Copyright

Copyright 1970, 1988 by Alfred S. Posamentier. All rights reserved under Pan American and International Copyright

Conventions.

Published in Canada by General Publishing Company, Ltd., 30 Lesmill Road, Don Mills, Toronto, Onlano.

Published in the United Kingdom by Constable and Company, Ltd., 3 The Lanchesters, 162-164 Fulham Palace Road. London W6 9ER.

Bibliographical Note

This Dover edition, first published in 1996, is an unabridged. very slightly altered republication of the work first published in 1970 by the Macmillan Com-pany, New York. and again in 1988 by Dale Seymour Publications. Palo Alto. California For the Dover edition, Professor Posamentier has made two slight alterations in the introductory material.

Library of Congress Cataloging-in-Publication Data

Posamentier. Alfred S. Challenging problems in algebra! Alfred S. Posamentier, Charles T Salkind

p. cm. "An unabridged, very slightly altered republication of the work first published

in 1970 by the Macmillan Company, New York. and again in 1988 by Dale Seymour Publications, Palo Alto, California. For the Dover edition, Professor Posamentier has made two slight alterations in the introductory material" -T.p verso.

ISBN 0-486-69148-9 (pbk ) I. Algebra-Problems, exercises, etc I Salkind. Charles T .

1898-. II. Title. QAI57.P68 1996 512.9'076-dc20 95-48215

CIP

Manufactured in the United States of Amenca Dover Publications, Inc., 31 East 2nd Street. Mineola. N.Y. 11501

CONTENTS

Introduction iv Preparing to Solve a Problem viii

SECTION I First Year Algebra

Problems Solutions 1. Posers: Innocent and Sophisticated 1 65 2. Arithmetic: Mean and Otherwise 6 74 3. Relations: Familiar and Surprising 10 84 4. Bases: Binary and Beyond 14 92 5. Equations, Inequations, and Pitfalls 17 99 6. Correspondence: Functionally Speaking 22 111 7. Equations and Inequations:

Traveling in Groups 25 121 8. Miscellaneous: Curiosity Cases 27 131

SECTION II Second Year Algebra

9. Diophantine Equations: The Whole Answer 30 139

10. Functions: A Correspondence Course 32 146 11. Inequalities, More or Less 35 152 12. Number Theory: Divide and Conquer 37 159 13. Maxima and Minima: Ups and Downs 41 173 14. Quadratic Equations: Fair and Square 43 180 15. Systems of Equations:

Strictly Simultaneous 45 185 16. Algebra and Geometry:

Often the Twain Shall Meet 47 191 17. Sequences and Series:

Progression Procession 51 201 18. Logarithms: A Power Play 57 220 19. Combinations and Probability:

Choices and Chances 58 222 20. An Algebraic Potpourri 61 229

Answers 240

Appendices 254

INTRODUCTION

The challenge of well-posed problems transcends national boundaries, ethnic origins, political systems, economic doctrines, and religious beliefs; the appeal is almost universal. Why? You are invited to formulate your own explanation. We simply accept the observation and exploit it here for entertainment and enrichment

This book is anew, combined edition of two volumes fIrSt published in 1970. It contains more than three hundred problems that are "off the beaten path"-problems that give a new twist to familiar topics or that introduce unfamiliar topics. With few exceptions, their solution requires little more than some knowledge of elementary algebra, though a dash of ingenuity may help. The problems range from fairly easy to hard, and many have extensions or variations that we call challenges. Supplied with pencil and paper and fortified with a diligent attitude, you can make this material the starting point for exploring unfamiliar or little-known aspects of mathematics. The challenges will spur you on; perhaps you can even supply your own challenges in some cases. A study of these non-routine problems can provide valuable underpinnings for work in more advanced mathematics.

This book. with slight modifications made, is as appropriate now as it was a quarter century ago when it was first published. The National Council of Teachers of Mathematics (NCTM), in their Curriculum and Evaluation Standards for High School Mathematics (1989), lists problem solving as its first standard, stating that "mathematical problem solving in its broadest sense is nearly synonymous with doing mathematics." They go on to say, "[problem solving] is a process by which the fabric of mathematics is identified in later standards as both constructive and reinforced. to

This strong emphasis on mathematics is by no means a new agenda item. In 1980, the NCTM published An Agendafor Action. There. the NCTM also had problem solving as its first item, stating, "educators should give priority to the identification and analysis of specific problem solving strate-gies .... [and] should develop and disseminate examples of 'good problems' and strategies." It is our intention to provide secondary mathematics educators with materials to help them implement this very important recommendation.

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ABOUT THE BOOK Challenging Problems in Algebra is organized into three main parts: "Problems," "Solutions," and "Answers." Unlike many contemporary problem-solving resources. this book is arranged not by problem-solving technique, but by topic. We feel that announcing the technique to be used stifles creativity and destroys a good part of the fun of problem solving.

The problems themselves are grouped into two sections. Section I covers eight topics that roughly parallel the sequence of a fust year algebra course. Section II presents twelve topics that roughly parallel the second year algebra course.

Within each topic, the problems are arranged in approximate order of difficulty. For some problems, the basic difficulty may lie in making the distinction between relevant and irrelevant data or between known and unknown information. The sure ability to make these distinctions is part of the process of problem solving, and each devotee must develop this power by him- or herself. It will come with sustained effort.

In the "Solutions" part of the book, each problem is restated and then its solution is given. From time to time we give alternate methods of solution, for there is rarely only one way to solve a problem. The solutions shown are far from exhaustive. and intentionally so, allowing you to try a variety of different approaches. Particularly enlightening is the strategy of using multiple methods. integrating algebra, geometry. and trigonometry. Instances of multiple methods or multiple interpretations appear in the solutions. Our continuing challenge to you, the reader. is to fmd a different method of solution for every problem.

The third part of the book, "Answers," has a double purpose. It contains the answers to all problems and challenges, providing a quick check when you have arrived at a solution. Without giving away the entire solution, these answers can also give you a hint about how to proceed when you are stuck.

Appendices at the end of the book provide information about several specialized topics that are not usually covered in the regular curriculum but are occasionally referred to in the solutions. This material should be of particular interest and merits special attention at the appropriate time.

THE TOPICS COVERED

Section I. The book begins with a chapter of general problems, simple to state and understand, that are generally appealing to students. These should serve as a pleasant introduction to problem solving early in the elementary algebra course.

Chapter 2 demonstrates the true value of algebra in understanding arithmetic phenomena. With the use of algebraic methods, students are guided through fascinating investigations of arithmetic curiosities.

vi

Familiar and unfamiliar relations are the bases for some cute problems in chapter 3. A refreshing consideration of various base systems is offered in chapter 4. Uncommon problems dealing with the common topics of equations, inequalities, functions, and simultaneous equations and inequalities are presented along with stimulating challenges in chapters 5, 6, and 7.

The last chapter of this section contains a collection of problems summarizing the techniques encountered earlier. These problems are best saved for the end of the elementary algebra course.

Section II. The second section opens with a chapter on one of the oldest forms of algebra, Diophantine equations-indeterminate equations for which only integer solutions are sought. These problems often appear formidable to the young algebra student, yet they can be solved easily after some experience with the type (which this section offers).

The next two chapters present some variations on familiar themes, functions and inequalities, treated here in a more sophisticated manner than was employed in the first section.

The field of number theory includes some interesting topics for the secondary school student, but all too often this area of study is avoided. Chapter 12 presents some of these concepts through a collection of unusual problems. Naturally, an algebraic approach is used throughout.

Aside from a brief exposure to maxium and minimum points on a parabola, very little is done with these concepts prior to a study of the calculus. Chapter 13 will demonstrate through problem solving some explorations of these concepts at a relatively elementary level.

Chapters 14, 15, 17, and 18 offer unconventional problems for some standard top