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More About This Textbook
Overview
A major aspect of mathematical training and its benefit to society is the ability to use logic to solve problems. The American Mathematics Competitions have been given for more than fifty years to millions of students. This book considers the basic ideas behind the solutions to the majority of these problems, and presents examples and exercises from past exams to illustrate the concepts. Anyone preparing for the Mathematical Olympiads will find many useful ideas here, but people generally interested in logical problem solving should also find the problems and their solutions stimulating. The book can be used either for selfstudy or as topicoriented material and samples of problems for practice exams. Useful reading for anyone who enjoys solving mathematical problems, and equally valuable for educators or parents who have children with mathematical interest and ability.
What People Are Saying
David Wells
"Within each chapter, three wellchosen examples illustrate a variety of problemsolving strategies and applications of concepts... The thoughtful choice of examples and exercises is one of the book's strengths, providing a wealth of opportunity for students to become experienced problem solvers within a remarkably small number of pages."—Penn State University
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Table of Contents
Preface; 1. Arithmetic ratios; 2. Polynomials and their zeros; 3. Exponentials and radicals; 4. Defined functions and operations; 5. Triangle geometry; 6. Circle geometry; 7. Polygons; 8. Counting; 9. Probability; 10. Prime decomposition; 11. Number theory; 12. Sequences and series; 13. Statistics; 14. Trigonometry; 15. Threedimensional geometry; 16. Functions; 17. Logarithms; 18. Complex numbers; Solutions to exercises; Epilogue; Sources of the exercises; Index; About the author.
Preface
In the last year of the second millennium, the American High School Mathematics Examination, commonly known as the AHSME, celebrated its fiftieth year. It began in 1950 as a local exam in the New York City area, but within its first decade had spread to most of the states and provinces in North America, and was being administered to over 150,000 students. A third generation of students is now taking the competitions.
The examination has expanded and developed over the years in a number of ways. Initially it was a 50question test in three parts. Part I consisted of 15 relatively routine computational problems; Part II contained 20 problems that required a thorough knowledge of high school mathematics, and perhaps some ingenuity; those in Part III were the most difficult, although some of these seem, based on the latter problems on the modern examination, relatively straightforward. The points awarded for success increased with the parts, and totaled 150. The exam was reduced to 40 questions in 1960 by deleting some of the more routine problems. The number of questions was reduced again, to 35, in 1968, but the number of parts was increased to four. The number of problems on the exam was finally reduced to 30, in 1974, and the division of the exam into parts with differing weights on each part was eliminated. After this time, each problem would be weighted equally. It continued in this form until the end of the century, by which time the exam was being given to over 240,000 students at over 5000 schools. One might get the impression that with a reduction in the number of problems the examination was becoming easier over the years, but a brief look at the earlier exams (which can be found The Contest Problem Book, Volumes I through V) will dissuade one from this view. The number of problems has been reduced, but the average level of difficulty has increased. There are no longer many routine problems on the exams, and the middlerange problems are more difficult than those in the early years. Since 1974, students from the United States have competed in the International Mathematical Olympiad (IMO), and beginning in 1972 students with very high scores on the AHSME were invited to take the United States of America Mathematical Olympiad (USAMO). The USAMO is a very difficult essaytype exam that is designed to select the premier problemsolving students in the country. There is a vast difference between the AHSME, a multiplechoice test designed for students with a wide range of abilities, and the USAMO, a test for the most capable in the nation. As a consequence, in 1983 an intermediate exam, the American Invitational Mathematics Examination was instituted, which the students scoring in approximately the top 5% on the AHSME were invited to take. Qualifying for the AIME, and solving even a modest number of these problems, quickly became a goal of many bright high school students, and was seen as a way to increase the chance of acceptance at some of the select colleges and universities. The plan of the top high school problem solvers was to do well enough on the AHSME to be invited to take the AIME, solve enough of the AIME problems to be invited to take the USAMO, and then solve enough USAMO problems to be chosen to represent the United States in the International Mathematical Olympiad. Also, of course, to do well in the IMO, that is, to win a Gold Medal! But I digress, back to the history of the basic exams. The success of the AHSME led in 1985 to the development of a parallel exam for middle school students, called the American Junior High School Mathematics Examination (AJHSME). The AJHSME was designed to help students begin their problemsolving training at an earlier age. By the end of the 20th century nearly 450,000 students were taking these exams, with representatives in each state and province in North America. In 2000 a major change was made to the AHSMEAJHSME system. Over the years there had been a reduction in the number of problems on the AHSME with a decrease in the number of relatively elementary problems. This reduction was dictated in large part by the demands of the school systems. Schools have had a dramatic increase in the number of both curricular and extracurricular activities, and time schedules are not as flexible as in earlier years. It was decided in 2000 to reduce the AHSME examination to 25 questions so that the exams could be given in a 75 minute period. However, this put students in the lower high school grades at an additional disadvantage, since it resulted in a further reduction of the more elementary problems. The Committee on the American Mathematics Competitions (CAMC) was particularly concerned that a capable student who had a bad experience with the exam in grades 9 or 10 might be discouraged from competing in later years. The solution was to revise the examination system by adding a competition specifically designed for students in grades 9 and 10. This resulted in three competitions, which were renamed AMC 8, AMC 10 and AMC 12. The digits following AMC indicate the highest grade level at which students are eligible to take the exam. There was no change in the AJHSME except for being renamed AMC 8, nor, except for the reduction in problems, was there a change in AHSME. The new AMC 10 was to consist of problems that could be worked with the mathematics generally taught to students in grades 9 and lower and there would be overlap, but not more than 50\%, between the AMC 10 and AMC 12 examinations. Excluded from the AMC 10 would be problems involving topics generally seen only by students in grades 11 and 12, including trigonometry, logarithms, complex numbers, functions, and some of the more advanced algebra and geometry techniques. The AMC 10 was designed so that students taking this competition are able to qualify for the AIME, however only approximately the top 1\% do so. The reason for making the qualifying score for AMC 10 students much higher than for AMC 12 students was threefold. First, there are students in grades 9 and 10 who have the mathematical knowledge required for the AMC 12, and these students should take the AMC 12 to demonstrate their superior ability. Having to score at the 1% level on the AMC 10 is likely to be seen to be riskier for these students than having to score at the 5% level on the AMC 12. Second, the committee wanted to be reasonably sure that a student who qualified for the AIME in grades 9 or 10 would also qualify when taking the AMC 12 in grades 11 and 12. Not to do so could discourage a sensitive student. Third, the AIME can be very intimidating to students who have not prepared for this type of examination. Although there has been a concerted effort recently to make the first group of problems on the AIME more elementary, there have been years when the median score on this 15question test was 0. It is quite possible for a clever 9th or 10th grader without additional training to do well on the AMC 10, but not be able to begin to solve an AIME problem. This, again, could discourage a sensitive student from competing in later years. The primary goal of the AMC is to promote interest in mathematics by providing a positive problemsolving experience for all students taking the exams. The AMC exam is also the first step in determining the top problemsolving high school students in the country, but that goal is decidedly secondary. My Experience with the American Mathematics Competitions My first formal involvement with the AMC began in 1996 when I was appointed to the CAMC as a representative from Pi Mu Epsilon, the National Honorary Mathematics Society. Simultaneously, I began writing problems for the AHSME and the AJHSME. In 1997 I joined the committee that constructs the examination for the AJHSME, based on problems submitted from a wide range of people in the United States and Canada. At the same time, I had been helping some local students in middle school prepare for the AJHSME and for the MathCounts competition, and had discovered how excited these students were even when they didn't do as well in the competitions as they had expected. The next year, when they were in 9th grade, I encouraged them to take the AHSME, since that was the only mathematical competition that was available to them. The level of difficulty on this AHSME was so much higher than the exams they were accustomed to taking that most of them were devastated by the experience. I believe that for all but two of these students this was their last competitive problemsolving experience. At the next meeting of the CAMC I brought my experience to the attention of the members and showed figures that demonstrated that only about 20% of the 9th grade students and less than 40% of the 10th grade students who had taken the AJHSME in grade 8 were taking the AHSME. Clearly, the majority of the 9th and 10th grade teachers had learned the lesson much earlier than I had, and were not encouraging their students to take the AHSME. At this meeting I proposed that we construct an intermediate exam for students in grades 9 and 10, one that would provide them with a better experience than the AHSME and encourage them to continue improving their problemsolving skills. As any experienced committee member knows, the person who proposes the task usually gets assigned the job. In 1999 Harold Reiter, the Director of the AHSME, and I became joint directors of the first AMC 10, which was first given on February 15, 2000.Since 2001 I have been the director of AMC 10. I work jointly with the AMC 12 director, Dave Wells, to construct the AMC 10 and AMC 12 exams. In 2002 we began to construct two sets of exams per year, the AMC 10A and AMC 12A, to be given near the beginning of February, and the AMC 10B and AMC 12B, which are given about two weeks later. This gives a student who has a conflict or unexpected difficulty on the day that the A version of the AMC exams are given a second chance to qualify for the AIME. For the exam committee, it means, however, that instead of constructing and refining 30 problems per year, as was done in 1999 for the AHSME, we need approximately 80 problems per year, 25 for each version of the AMC 10 and AMC 12, with an overlap of approximately ten problems. There are a number of conflicting goals associated with constructing the A and B versions of the exams. We want the versions of the exams to be comparable, but not similar, since similarity would give an advantage to the students taking the later exam. Both versions should also contain the same relative types of problems, but be different, so as not to be predictable. Additionally, the level of difficulty of the two versions should be comparable, which is what we have found most difficult to predict. We are still in the process of grappling with these problems but progress, while slow, seems to be steady. The Basis and Reason for this Book When I became a member of the Committee on American Competitions, I found that students in the state of Ohio had generally done well on the exams, but students in my local area were significantly less successful. By that time I had over 25 years experience working with undergraduate students at Youngstown State University and, although we had not done much with problemsolving competitions, our students had done outstanding work in undergraduate research presentations and were very competitive on the international mathematical modeling competition sponsored by COMAP. Since most of the Youngstown State students went to high school in the local area, it appeared that their performance on the AHSME was not due to lack of ability, but rather lack of training. The mathematics and strategies required for successful problem solving is not necessarily the same as that required in general mathematical applications. In 1997 we began to offer a series of training sessions at Youngstown State University for high school students interested in taking the AHSME, meeting each Saturday morning from 10:00 until 11:30. The sessions began at the end of October and lasted until February, when the AHSME was given. The sessions were attended by between 30 and 70 high school students. Each Saturday about three YSU faculty, a couple of very good local high school teachers, and between five and ten YSU undergraduate students presented some topics in mathematics, and then helped the high school students with a collection of exercises. The first year we concentrated each week on a specific past examination, but this was not a successful strategy. We soon found that the variability in the material needed to solve the problems was such that we could not come close to covering a complete exam in the time we had available. Beginning with the 19981999 academic year, the sessions were organized by mathematical topic. We used only past AHSME problems and found a selection in each topic area that would fairly represent the type of mathematical techniques needed to solve a wide range of problems. The AHSME was at that time a 30question exam and we concentrated on the problem range from 6 to 25. Our logic was that a student who could solve half the problems in this range could likely do all the first five problems and thus easily qualify for the AIME. Also, the last few problems on the AHSME are generally too difficult to be accessible to the large group we were working with in the time we had available. This book is based on the philosophy of sessions that were run at Youngstown State University. All the problems are from the past AMC (or AHSME, I will not subsequently distinguish between them) exams. However, the problems have been edited to conform with the modern mathematical practice that is used on current AMC examinations. So, the ideas and objectives of the problems are the same as those on past exams, but the phrasing, and occasionally the answer choices, have been modified. In addition, all solutions given to the Examples and the Exercises have been rewritten to conform to the material that is presented in the chapter. Sometimes this solution agrees with the official examination solution, sometimes not. Multiple solutions have occasionally been included to show students that there is generally more than one way to approach the solution to a problem. The goal of the book is simple. To promote interest in mathematics by providing students with the tools to attack problems that occur on mathematical problemsolving exams, and specifically to level the playing field for those who do not have access to the enrichment programs that are common at the top academic high schools. The material is written with the assumption that the topic material is not completely new to the student, but that the classroom emphasis might have been different. The book can be used either for self study or to give people who would want to help students prepare for mathematics exams easy access to topicoriented material and samples of problems based on that material. This should be useful for teachers who want to hold special sessions for students, but it should be equally valuable for parents who have children with mathematical interest and ability. One thing that we found when running our sessions at Youngstown State was that the regularly participating students not only improved their scores on the AMC exams, but did very well on the mathematical portion of the standardized college admissions tests. (No claim is made concerning the verbal portion, I hasten to add.) I would like to particularly emphasize that this material is not a substitute for The Contest Problem Book, Volumes I through VIII. Those books contain multiple approaches to solutions to the problems as well as helpful hints for why particular ``foils'' for the problems were constructed. My goal is different, I want to show students how a few basic mathematical topics can be used to solve a wide range of problems. I am using the AMC problems for this purpose because I find them to be the best and most accessible resource to illustrate and motivate the mathematical topics that students will find useful in many problemsolving situations. Finally, let me make clear that the student audience for this book is perhaps the top 1015\% of an average high school class. The book is not designed to meet the needs of elite problem solvers, although it might give them an introduction that they might otherwise not be able to find. References are included in the Epilogue for more advanced material that should provide a challenge to those who are interested in pursuing problem solving at the highest level. Structure of the Book Each chapter begins with a discussion of the mathematical topics needed for problem solving, followed by three Examples chosen to illustrate the range of topics and difficulty. Then there are ten Exercises, generally arranged in increasing order of difficulty, all of which have been on past AMC examinations. These Exercises contain problems ranging from relatively easy to quite difficult. The Examples have detailed solutions accompanying them. The Exercises also have solutions, of course, but these are placed in a separate Solutions chapter near the end of the book. This permits a student to read the material concerning a topic, look at the Examples and their solutions, and then attempt the Exercises before looking at the solutions that I have provided.Within the constraints of wide topic coverage, problems on the most recent examinations have been chosen. It is, I feel, important to keep in mind that a problem on an exam as recent as 1990 was written before many of our current competitors were born! The first four chapters contain rather elementary material and the problems are not difficult. This material is intended to be accessible to students in grade 9. By the fifth chapter on triangle geometry there are some more advanced problems. However, triangle geometry is such an important subject on the examinations, that there are additional problems involving these concepts in the circle geometry and polygon chapters. Chapters 8 and 9 concern counting techniques and probability problems. There is no advanced material in these chapters, but some of the probability problems can be difficult. More counting and probability problems are considered in later chapters. For example, there are trigonometry and threedimensional geometry problems that require these notions. Chapters 10 and 11 concern problems with integer solutions. Since these problems frequently occur on the AMC, Chapter 10 is restricted to those problems that essentially deal with the Fundamental Theorem of Arithmetic, whereas Chapter 11 considers the more advanced topics of modular arithmetic and number bases. All of this material should be accessible to an interested younger student. Chapter 12 deals with sequences and series, with an emphasis on the arithmetic and geometric sequences that often occur on the AMC. Sequences whose terms are recursive and repeat are also considered, since the AMC sequence problems that are not arithmetic or geometric are frequently of this type. This material and that in Chapter 13 that deals with statistics may not be completely familiar to younger students, but there are only a few concepts to master, and some of these problems appear on the AMC 10. The final four chapters contain material that is not likely to be included on an AMC 10. Definitions for the basic trigonometric and logarithm functions are given in Chapters 14 and 17, respectively, but these may not be sufficient for a student who has not previously seen this material before. Chapter 15 considers problems that have a threedimensional slant, and Chapter 16 looks at functions in a somewhat abstract setting. The final chapter on complex numbers illustrates that the knowledge of just a few concepts concerning this topic is all that is generally required, even for the AMC 12. One of the goals of the book is to permit a student to progress through the material in sequence. As problemsolving abilities improve, more difficult notions can be included, and problems presented that require greater ingenuity. When reviewing this material I hope that you will keep in mind that the intended student audience for this book is perhaps the top 1015\% of an average high school class. The more mature (think parental) audience is probably the working engineer or scientist who has not done problems of this type for many years, if ever, but enjoys a logical challenge and/or wants to help students develop problem solving skills.