陶哲轩教你学数学

Solving Mathematical Problems: A Personal Perspective Terence Tao Department of Mathematics, UCLA, Los Angeles, CA 90095 E-mail address: tao@math.ucla.edu Dedicated to all my mentors, who taught me the meaning (and joy) of mathematics. Contents Foreword to the first edition ix Preface to the first edition xi Preface to the second edition xv Chapter 1. Strategies in problem solving 1 Chapter 2. Examples in number theory 9 Chapter 3. Examples in algebra and analysis 31 Chapter 4. Euclidean geometry 43 Chapter 5. Analytic geometry 63 Chapter 6. Sundry examples 77 References 91 vii Foreword to the first edition This is Terry Tao’s first book. The manuscript was prepared early in 1991, when Terry was 15 years of age. We, at Deakin University, commisioned Terry to write a book on mathematical problem solving which would be suitable for use in a Deakin University course taken mainly by practising school teachers. The brief given to Terry was to write a book that would be at least partly com- prehensible to those who did not have high formal mathematical qualifications, yet would enable all readers, whatever their mathematical backgrounds, to appreciate the beauty of elegant problem-solving stragies. The outcome of Terry’s effort is a work which, we are confident, will inspire teachers and students of mathematics at all levels to reflect on the obvious youthful zest, joy, yet dogged determiniation to achieve an excellent result, that characterise Terry’s responses to challenging mathematical problems. Since it was to be Terry’s first book, we wanted it to be a work which, in the future, he would regard as something special. Given Terry’s mathematical precocity we realised, of course, that it was likely that the book would find its place on many school and college library shelves around the world, and we wanted it to stand as a vibrant testimony to how an outstanding mathematical mind went about solving challenging mathematical problems. Clearly, the instructions we gave Terry defined a highly problematic task. How could anyone write a book that revealed deep (yet, somewhat paradoxically, ap- parently simple) mathematical insights but which was simultaneously capable of being appreciated (if not fully understood) by persons without large and formal mathematical backgrounds? You, the reader, will be the judge of how well this problem is solved in this book. The Author The interested reader is referred to published articles on Terry Tao (Clements 1984; Gross 1986) for more complete biographical details than can be provided here. Terry was born in Adelaide in July 1975, the eldest son of Billy (a pediatrician) and Grace (an honours graduate in physics and mathematics). His parents recognized quite early that he had mathematical talent and in 1983, aged 7, he was allowed to study mathematics at a local high school. At the end of 1983 he passed the ix x FOREWORD TO THE FIRST EDITION South Australian matriculation Mathematics 1 and Mathematics 2 examinations with scores of 90% and 85% respectively, and in 1984, aged 8 years, he scored 760 on the mathematical portion of the College Board (USA) Scholastic Aptitude Test (SAT-M), a result higher than any that had been achieved by a North American child of the same age, and one that was above 99th percentile for college-bound, 12th-graders in the United States. In 1986, 1987, and 1988, Terry obtained bronze, silver, and gold medals, respec- tively, for Australia in the International Mathematics Olympiad. He obtained the ‘gold’ during the month he turned 13, and is easily the youngest gold medal win- ner form any country in the history of the Olympiads. In 1989 he enrolled as a full-time student at Flinders University (in Adelaide), and in December 1990 he completed his BSc degree at Flinders, receiving a special letter of commendation from the Chancellor. In 1991 he completed a BSc honours degree in mathematics, at Flinders, and in 1992, aged 17, he commenced PhD studies in mathematics at Princeton University in the United States. In 1986 Miraca Gross said of Terry, ‘He is a delightful young boy who is aware that he is different but displays no conceit about his remarkable gifts and has an unusual ability to relate to a wide range of people, from children younger than himself to the university faculty members’ (Gross 1986, p. 5). As someone who as watched Terry’s development over the years, I can say that the same comment still applies today. Throughout the pages of this book you will discover an impish, yet subtle sense of humour that interacts, in an intguiging way, with an obvious and overwhelming desire to achieve the best possible solution. Julian Stanley, the Johns Hopkins University professor who, for many years, has studied mathematically precocious youngsters in the United States, was moved to write that sometimes he thought that he and his colleagues ‘were learning more from Terry than he and his parents were learning from us’ (Stanley 1986, p. 11). Stanley’s comments are relevant in a broader educational context - this book has something to teach us all. I believe that all persons interested in mathematics, and even many who do not profess such an interest, will, by reading this book, be challenged to reflect on many general educational issues - not least of which is the question of what our schools are doing, and what they could be doing, to meet the interests and needs of those with special gifts. References (1) Clements, M.A. (1984), Terence Tao, Educational Studies in Mathematics 13, 213–238. (2) Gross, M. (1986), Radical acceleration in Australia: Terence Tao, G/C/T 9(1), 2–9. (3) Staney, J.C. (1986), Insights, G/C/T 9(1), 10–11. M.A. (Ken) Clements Faculty of Education Deakin University December 1991 Preface to the first edition Proclus , an ancient Greek philosopher, said: This therefore, is mathematics: she reminds you of the invisible forms of the soul; she gives life to her own discoveries; she awakens the mind and purifies the intellect; she brings to light our intrinsic ideas; she abolishes oblivion and ignorance which are ours by birth . . . But I just like mathematics because it’s fun. Mathematical problems, or puzzles, are important to real mathematics (like solving real-life problems), just as fables, stories and anecdotes are important to the young in understanding real life. Mathematical problems are “sanitized” mathematics, where an elegant solution has already been found (by someone else, of course), the question is stripped of all superfluousness and posed in an interesting and (hopefully) thought-provoking way. If mathematics is likened to prospecting for gold, solving a good mathematical problem is akin to a “hide-and-seek” course in gold-prospecting: you are given a nugget to find, and you know what it looks like, that it is out there somewhere, that it is not too hard to reach, that unearthing it is within your capabilities, and you have conveniently been given the right equipment (i.e. data) to get it. It may be hidden in a cunning place, but it will require ingenuity rather than digging to reach it. In this book I shall solve selected problems from various levels and branches of mathematics. Starred problems (*) indicate an additional level of difficulty, either because some higher mathematics or some clever thinking are required; double- starred questions (**) are similar, but to a greater degree. Some problems have additional exercises at the end that can be solved in a similar manner or involve a similar piece of mathematics. While solving these problems, I will try to demon- strate some tricks of the trade when problem-solving. Two of the main weapons - experience and knowledge - are not easy to put into a book: they have to be acquired over time. But there are many simpler tricks that take less time to learn. There are ways of looking at a problem that make it easier to find a feasible attack plan. There are systematic ways of reducing a problem into successively simpler sub-problems. But, on the other hand, solving the problem is not everything. To return to the gold nugget analogy, strip-mining the neighbourhood with bulldoz- ers is clumsier than doing a careful survey, a bit of geology, and a small amount of digging. A solution should be relatively short, understandable, and hopefully have a touch of elegance. It should also be fun to discover. Transforming a nice, xi xii PREFACE TO THE FIRST EDITION short little geometry question into a ravening monster of an equation by textbook coordinate geometry doesn’t have the same taste of victory as a two-line vector solution. As an example of elegance, here is a standard result in Euclidean geometry: Show that the perpendicular bisectors of a triangle are concurrent. This neat little one-liner could be attacked by coordinate geometry. Try to do so for a few minutes (hours?), then look at this solution: A C B P Proof. Call the triangle ABC. Now let P be the intersection of the perpen- dicular bisectors of AB and AC. Because P is on the AB bisector, |AP | = |PB|. Because P is on the AC bisector, |AP | = |PC|. Combining the two, |BP | = |PC|. But this means that P has to be on the BC bisector. Hence all three bisectors are concurrent. (Incidentally, P is the circumcentre of ABC.) � The following reduced diagram shows why |AP | = |PB| if P is on the AB perpen- dicular bisector: congruent triangles will pull it off nicely. A B P This kind of solution - and the strange way that obvious facts mesh to form a not-so-obvious fact - is part of the beauty of mathematics. I hope that you too will appreciate this beauty. Acknowledgements Thanks to Peter O’Halloran, Vern Treilibs, and Lenny Ng for their contributions of problems and advice. ACKNOWLEDGEMENTS xiii Special thanks to Basil Rennie for corrections and ingenious shortcuts in solutions, and finally thanks to my family for support, encouragement, spelling corrections, and put-downs when I was behind schedule. Almost all of the problems in this book come from published collections of problem sets for mathematics competitions. These are sourced in the texts, with full details given in the reference section of the book. I also used a small handful of problems from friends or from various mathematical publications; these have no source listed. Preface to the second edition This book was written fifteen years ago; literally half a lifetime ago, for me. In the intervening years, I have left home, moved to a different country, gone to graduate school, taught classes, written research papers, advised graduate students, married my wife, and had a son. Clearly, my perspective on life and on mathematics is different now than it was when I was fifteen; I have not been involved in problem- solving competitions for a very long time now, and if I were to write a book now on the subject it would be very different from the one you are reading here. Mathematics is a multifaceted subject, and our experience and appreciation of it changes with time and experience. As a primary school student, I was drawn to mathematics by the abstract beauty of formal manipulation, and the remarkable ability to repeatedly use simple rules to achieve non-trivial answers. As a high- school student, competing in mathematics competitions, I enjoyed mathematics as a sport, taking cleverly designed mathematical puzzle problems (such as those in this book) and searching for the right “trick” that would unlock each one. As an undergraduate, I was awed by my first glimpses of the rich, deep, and fascinating theories and structures which lie at the core of modern mathematics today. As a graduate student, I learnt the pride of having one’s own research project, and the unique satisfaction that comes from creating an original argument that resolved a previously open question. Upon starting my career as a professional research mathematician, I began to see the intuition and motivation that lay behind the theories and problems of modern mathematics, and was delighted when realizing how even very complex and deep results are often at heart be guided by very simple, even common-sensical, principles. The “Aha!” experience of grasping one of these principles, and suddenly seeing how it illuminates and informs a large body of mathematics, is a truly remarkable one. And there are yet more aspects of mathematics to discover; it is only recently for me that I have grasped enough fields of mathematics to begin to get a sense of the endeavour of modern mathematics as a unified subject, and how it connects to the sciences and other disciplines. As I wrote this book before my professional mathematics career, many of these insights and experiences were not available to me, and so the writing here is when I wrote this book, and so in many places the exposition has a certain innocence, or even naivete. I have been reluctant to tamper too much with this, as my younger self was almost certainly more attuned to the world of the high-school problem solver than I am now. However, I have made a number of organizational changes, arranging the material into what I believe is a more logical order, and editing those xv xvi PREFACE TO THE SECOND EDITION parts of the text which were inaccurate, badly worded, confusing, or unfocused. I have also added some more exercises. In some places, the text is a bit dated (Fermat’s last theorem, for instance, has now been proved rigourously), and I now realize that several of the problems here could be handled more quickly and cleanly by more “high-tech” mathematical tools; but the point of this text is not to present the slickest solution to a problem or to provide the most up-to-date survey of results, but rather to show how one approaches a mathematical problem for the first time, and how the painstaking, systematic experience of trying some ideas, eliminating others, and steadily manipulating the problem can lead, ultimately, to a satisfying solution. I am greatly indebted to Tony Gardiner for encouraging and supporting the reprint- ing of this book, and to my parents for all their support over the years. I am also touched by all the friends and acquaintances I have met over the years who had read the first edition of the book. Last, but not least, I owe a special debt to my parents and the Flinders Medical Centre computer support unit for retrieving a fifteen-year old electronic copy of this book from our venerable Macintosh Plus computer! Terence Tao Department of Mathematics, University of California, Los Angeles December 2005 CHAPTER 1 Strategies in problem solving The journey of a thousand miles begins with one step. - Lao Tzu Like and unlike the proverb above, the solution to a problem begins (and continues, and ends) with simple, logical steps. But as long as one steps in a firm, clear direction, with long strides and sharp vision, one would need far, far less than the millions of steps needed to journey a thousand miles. And mathematics, being abstract, has no physical constraints; one can always restart from scratch, try new avenues of attack, or backtrack at an instant’s notice. One does not always have these luxuries in other forms of problem-solving (e.g. trying to go home if you are lost). Of course, this does not necessarily make it easy; if it was easy, then this book would be substantially shorter. But it makes it possible. There are several general strategies and perspectives to solve a problem correctly; (Polya, 1948) is a classic reference for many of these. Some of these strategies are discussed below, together with a brief illustration of how each strategy can be used on the following problem: Problem 1.1. A triangle has its lengths in an arithmetic progres- sion, with difference d. The area of the triangle is t. Find the lengths and angles of the triangle. Understand the problem. What kind of problem is it? There are three main types of problems: • “Show that . . . ” or “Evaluate . . . ” questions, in which a certain statement has to be proved true, or a certain expression has to be worked out; • “Find a . . . ” or “Find all . . . ” questions, which requires one to find something (or everything) that satisfies certain requirements; and • “Is there a . . . ” questions, which either require you to prove a statement or provide a counterexample (and thus is one of the previous two types of problem). The type of problem is important because it determines the basic method of ap- proach. “Show that . . . ” or “Evaluate . . . ” problems start with given data and the 1 2 1. STRATEGIES IN PROBLEM SOLVING objective is to deduce some statement or find the value of an expression; this type of problem is generally easier than the other two types because there is a clearly visible objective, one that can be deliberately approached. “Find a . . . ” questions are more hit-and-miss; generally one has to guess one answer that nearly works, and then tweak it a bit to make it more correct; or alternatively one can alter the requirements that the object-to-find must satisfy, so that they are easier to satisfy. “Is there a . . . ” problems are typically the hardest, because one first must make a decision on whether an object exists or not, and provide a proof on one hand, or a counter-example on the other. Of course, not all questions fall into these neat categories; but the general format of any question will still show the basic idea to pursue when solving a problem. For example, if one tries to solve the problem “Find a hotel in this city to sleep in for the night”, one should alter the requirements to, say “Find a vacant hotel within 5 kilometres with a room that costs less than 100$ a night” and then use pure elimination. This is a better strategy than proving that such a hotel does or does not exist, and is probably a better strategy than picking any handy hotel and trying to prove that one can sleep in it. In Problem 1.1 question, we have an “Evaluate. . . ” type of problem. We need to find several unknowns, given other variables. This suggests an algebraic solution rather than a geometric one, with a lot of equations connecting d, t, and the sides and angles of the triangle, and eventually solving for our unknowns. Understand the data. What is given in the problem? Usually, a question talks about a number of objects which satisfy some special requirements. To un- derstand the data, one needs to see how the objects and requirements react to each other. This is important in focusing attention on the proper techniques and nota- tion to handle the problem. For example, in our sample question, our data are a triangle, the area of the triangle, and the fact that the sides are in an arithmetic progression with separation d. Because we have a triangle, and are considering the sides and area of it, we would need theorems relating sides, angles, and areas to tackle the question: the sine rule, cosine rule, and the area formulas, for exam- ple. Also, we are dealing with an arithmetic progression, so we would need some notation to account for that; for example, the side lengths could be a, a + d, and a + 2d. Understand the objective. What do we want? One may need to find an ob- ject, prove a statement, determine the existence of a object with special properties, or whatever. Like the flip side of this strategy, “Understand the data”, knowing the objective helps focus attention on the best weapons to use. Knowing the ob- jective also helps in creating tactical goals which we know will bring us closer to solving the question. Our example question has the objective of “Find all the sides and angles of the triangle”. This means, as mentioned before, that we will need theorems and results concerning sides and angles. It also gives us the tactical goal of “find equations involving the sides and angles of the triangle”. 1. STRATEGIES IN PROBLEM SOLVING 3 Select good notation. Now that we have our data and objective,