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First Steps for Math Olympians

First Steps for Math Olympians: Using the American Mathematics Competitions

J. Douglas Faires
Series: Problem Books
Copyright Date: 2006
Edition: 1
Pages: 330
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  • Book Info
    First Steps for Math Olympians
    Book Description:

    A major aspect of mathematical training and its benefit to society is the ability to use logic to solve problems. The American Mathematics Competitions (AMC) have been given for more than 50 years to millions of high school 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 taking the AMC exams or helping students prepare for them will find many useful ideas here. This book will promote interest in mathematics by providing students with the tools to attack problems that occur on mathematical problem-solving 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 book can be used either for self-study or to give people who want to help students prepare for mathematics exams easy access to topic-oriented material and samples of problems based on that material. This is useful for teachers who want to hold special sessions for students, but it is equally valuable for parents who have children with mathematical interest and ability.

    eISBN: 978-1-61444-404-6
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. Table of Contents
    (pp. vii-xii)
  3. Preface
    (pp. xiii-xxii)
    Doug Faires
  4. 1 Arithmetic Ratios
    (pp. 1-8)

    Nearly every AMC exam contains problems that require no more mathematical knowledge than the manipulation of fractions and ratios. The most difficult aspect of these problems is translating information given in sentences into an equation form.

    Problems involving time, distance, and average rates of speed are popular because the amount of knowledge needed to solve the problem is minimal, simply that

    Distance = Rate = Time.

    However, the particular phrasing of the problem determines how this formula should be used. Consider the following:

    Problem 1 You drive for one hour at 60 mph and then drive one hour at 40...

  5. 2 Polynomials and their Zeros
    (pp. 9-18)

    The most basic set of numbers is the integers. In a similar manner, the most basic set of functions is the polynomials. Because polynomials have so many applications and are relatively easy to manipulate, they appear in many of the problems on the AMC.

    Definition 1 A polynomial of degreenhas the form\[P(x)={{a}_{n}}{{x}^{n}}+{{a}_{n-1}}{{x}^{n-1}}+\cdot \cdot \cdot +{{a}_{1}}x+{{a}_{0}}\]for some collection of constantsa0,a1, … ,an, withleading coefficient an≠ 0. We will assume these constants are all real numbers.

    One of the most commonly needed features of a polynomial is the location of those values ofxsuch that...

  6. 3 Exponentials and Radicals
    (pp. 19-28)

    There is little to learn in order to work the exponential problems on the AMC. It is mainly a matter of knowing and applying the relevant definitions and arithmetic properties. However, this topic always seems to cause some students difficulty. As a consequence, most AMC exams include problems that use this knowledge.

    The first step is to develop working definitions of exponentials.

    Ifnis a positive integer andais a real number, then the symbol\[{{a}^{n}}=\overbrace{a\cdot a\cdot \cdot \cdot a}^{n-times}\]represents the product ofnfactors ofa. In the expressionan, the numberais called the base andnis...

  7. 4 Defined Functions and Operations
    (pp. 29-36)

    Problems on the AMC that use the material in this chapter are primarily manipulative. Often their solution requires only a careful application of a definition that may not be familiar, but is not difficult to comprehend.

    A binary operation on a set of numbers is simply a way to take two of the numbers and produce a third. Quite often a binary operation in an exam problem will be expressed using some unusual symbol, such as §. The result of the operation after it is applied to the numbersaandbwould likely be written as §(a,b), or...

  8. 5 Triangle Geometry
    (pp. 37-54)

    Every AMC has included problems on triangle geometry, generally in the medium to relatively difficult range. The higher-range problems often require some geometric construction to obtain the solution. Although the subject matter necessary to solve these problems is generally included in a standard high school geometry course, the emphasis in the course may not be sufficient for the solution of all of these problems.

    We will assume the basic concepts of geometry, such as the definition of a point, a line, an angle, and so on, are known. Our starting point will be the triangle, a geometric figure in the...

  9. 6 Circle Geometry
    (pp. 55-70)

    This chapter continues the subject of geometry in the plane. There are many types of problems that use circles in their solution, some involving triangles as well as circles. Many of the problems that involve circles are most easily solved using equations to represent that circle, but these will be postponed to a later chapter. Here we consider only those problems that strictly involve plane geometry.

    There are numerous definitions and results in this material, and it is important to have complete familiarity with the notation.

    We begin with the basic definitions and include here all the terminology that will...

  10. 7 Polygons
    (pp. 71-84)

    In Chapter 5 we considered the geometric properties of triangles, whose sides are composed of three straight line segments. In this chapter we expand the topic to more general geometric figures whose sides are straight line segments. These are calledpolygons.

    Definition 1 A polygon is a geometric figure in the plane whose sides consist of straight line segments, and no two consecutive sides lie on the same straight line.

    Definition 2 Ann-sided polygon is called ann-gon. The most commonn-gons have special names:

    A 3-gon is a triangle;

    A 4-gon is a quadrilateral;

    A 5-gon is a...

  11. 8 Counting
    (pp. 85-96)

    This chapter considers problems that involve permutations, combinations, partitioning, and other counting-oriented problems. Some AMC problems involve nothing more than the application of these ideas, others use these counting techniques as a first step when solving a more complicated problem.

    Definition 1 A permutation of a collection of distinguishable objects is an arrangement of the objects in some specific order.

    For example,acbdanddabcare both permutations of the lettersa,b,c, andd. What generally interests us is the number of different permutations that are possible from a given collection. In this case there are 24 different...

  12. 9 Probability
    (pp. 97-108)

    Nearly every AMC examination involves problems on probability. These problems often incorporate the notions of permutations, combinations, and other counting techniques. Probability problems are sometimes difficult to interpret and a careful reading of the problem is most important.

    The basic notions of probability that appear in problems on the AMC examinations involve situations where it is necessary to count the number of possible successes of some specified outcome as well as the number of all possible outcomes. When the events are equally likely to occur, the probability of success is the quotient of these two numbers.

    Definition 1 The probability...

  13. 10 Prime Decomposition
    (pp. 109-114)

    This first chapter on number theory topics considers problems that use the Fundamental Theorem of Arithmetic, which states that every positive integer has a unique prime factorization. Many AMC problems use this decomposition as a first step in finding the number of ways that a product can be factored or that certain things can occur.

    The positive integers, ornatural numbers, are the fundamental building blocks of arithmetic, and the prime numbers form the basis for the natural numbers.

    Definition 1 A natural number greater than 1 is said to be prime if its only natural number divisors are 1...

  14. 11 Number Theory
    (pp. 115-126)

    Number Theory is a subject used to describe a multitude of different types of problems, with the major commonality being that the solutions to these problems are integers. The study began in the previous chapter on Prime Decomposition, but problems that have integer solutions are sufficiently common and varied to require a chapter of their own.

    Some of the problems considered in this section involve the expression of numbers in bases other than 10. These should not cause any difficulty if we keep in mind how our common base-10 representation is defined. A common base-10 number written, for example, in...

  15. 12 Sequences and Series
    (pp. 127-134)

    This chapter considers the common arithmetic and geometric sequences and series, as well as sequences and series that are defined inductively and recursively. Pattern recognition often plays a major role in the solution of these problems.

    By a sequence of numbers we simply mean an ordered way of writing the numbers. To be more precise, we use the notion of a function.

    Definition 1 A sequence of numbers is a function that assigns to each positive integer a distinct number. The number assigned by the sequence to the integernis commonly denoted using a subscript, such asan. These...

  16. 13 Statistics
    (pp. 135-142)

    Most of the AMC statistics problems involve the concepts of mean, median, and mode. These problems are not generally difficult, but the concepts might not be familiar to all students. Students taking the AMC 10 will not likely see problems involving statistics, unless the definitions of the statistical concepts are given in the problem.

    Statistics problems can also involve graph interpretation, as well as counting methods such as permutations and combinations.

    Suppose that we are given a collection of numbers, {a1,a2, … ,an}. There are various ways to describe the way in which these numbers are distributed.


  17. 14 Trigonometry
    (pp. 143-154)

    This chapter begins the consideration of mathematical topics expected to be known for the AMC 12 exam but not for the AMC 10. There have not been many problems involving trigonometry on the more recent exams because the use of calculators makes many of the traditional problems trivial. However, the topic is important and the subject matter dealing with this subject is quite general. Students taking the AMC 10 examinations will not see problems involving trigonometry.

    The two very basic definitions in trigonometry are the sine and the cosine of a given number or given angle. There are two standard...

  18. 15 Three-Dimensional Geometry
    (pp. 155-166)

    Many AMC exams have some three-dimensional problems among the latter offerings. Often these problems are not difficult, but involve topics that are unfamiliar to many students. In this chapter the properties of standard three-dimensional objects are considered, and it is shown how these concepts are derived from more familiar planar objects. Students taking the AMC 10 are less likely to see problems involving three-dimensional geometry.

    Many of the problems involving 3 dimensions require little more than the ability to visualize in three-dimensional space. To do this more easily, it is common practice to use a perspective view of objects in...

  19. 16 Functions
    (pp. 167-178)

    The properties of functions are now well known to juniors and seniors in high school, and the recent AMCs have included increasing numbers of problems associated with functions and functional notation. Many of these problems require only careful manipulation of the functional definitions.

    Students taking the AMC 10 will not likely see problems involving the function concepts considered in this chapter.

    A function is a specific way to associate the elements of one set with the elements of another set.

    Definition 1 A function from setAto setBis a means of associating every element of the set...

  20. 17 Logarithms
    (pp. 179-186)

    Logarithm properties often cause students difficulty, but the number of concepts to master in order to do these problems is rather small. As a consequence, even though most problems involving logarithms come later in the exams, they are often not difficult. This is another topic that would be more heavily emphasized if calculators were excluded from the exam. Students taking the AMC 10 examinations will not see problems involving logarithms.

    Definition I Ifa≠ 1 andxare positive real numbers, then we say thatyis the logarithm ofxto the basea, writteny= logax,...

  21. 18 Complex Numbers
    (pp. 187-196)

    Most of the AMC 12 examinations include a problem that deals with complex numbers. These problems generally come near the end of the examination, since many contestants are not familiar with this subject and the manipulations can be intricate. However, there are just a few concepts needed to solve these problems, so they can sometimes be relatively easy. Students taking the AMC 10 examinations will not see problems involving complex numbers.

    Definition 1 The set of complex numbers, denoted${\mathcal C}$, consists of all expressions of the formz=a+bi, whereaandbare real numbers...

  22. Solutions to Exercises
    (pp. 197-290)
  23. Epilogue
    (pp. 291-294)
    Doug Faires

    This brief section contains some references to problem-solving material for students on the high school level that either supplements what I have given here or goes beyond the techniques I have described. There is a vast amount of problem-solving material, and if you are seriously interested in this activity you will no doubt find a favorite that I have not included here. These are simply a few of my personal favorites. The goal is not to be inclusive, but simply to provide paths for you to begin your journey.

    Many of the books I will list are published by the...

  24. Sources of the Exercises
    (pp. 295-300)
  25. Index
    (pp. 301-306)
  26. About the Author
    (pp. 307-307)