The Art of Problem Solving, Volume 1: the Basics

The Art of Problem Solving, Volume 1 , is the classic problem solving textbook used by many successful MATHCOUNTS programs, and has been an important building block for students who, like the authors, performed well enough on the American Mathematics Contest series to qualify for the Math Olympiad Summer Program which trains students for the United States International Math Olympiad team. Volume 1 is appropriate for students just beginning in math contests. MATHCOUNTS and novice high school students particularly have found it invaluable. Although the Art of Problem Solving is widely used by students preparing for mathematics competitions, the book is not just a collection of tricks. The emphasis on learning and understanding methods rather than memorizing formulas enables students to solve large classes of problems beyond those presented in the book. Speaking of problems, the Art of Problem Solving, Volume 1 , contains over 500 examples and exercises culled from such contests as MATHCOUNTS, the Mandelbrot Competition, the AMC tests, and ARML. Full solutions (not just answers!) for all the problems are built into the book. Title page Copyright page Thanks To Students To Teachers Contents 1. Exponents and Logarithms 1.1. Integer Exponents 1.2. Fractional Exponents 1.3. Simplifying Radical Expressions 1.4. Rationalizing Denominators 1.5. Logarithms Problems to Solve for Chapter 1 the BIG PICTURE 2. Complex Numbers 2.1. The Square Root of -1 2.2. Complex Number Operations Problems to Solve for Chapter 2 3. Linear Equations 3.1. What is a Linear Equation? 3.2. One Equation, One Variable 3.3. Two Equations, Two Variables 3.4. Word Problems Problems to Solve for Chapter 3 4. Proportions 4.1. Direct and Inverse 4.2. Manipulating Proportions 4.3. Conversion Factors 4.4. Percent Problems to Solve for Chapter 4 the BIG PICTURE 5. Using the Integers 5.1. Divisibility 5.2. Number Bases 5.3. The Last Digit 5.4. Modular Arithmetic 5.5. Tricks 5.6. Primes 5.7. Common and Uncommon Factors Problems to Solve for Chapter 5 the BIG PICTURE 6. Quadratic Equations 6.1. What's a Quadratic? 6.2. Factoring Quadratics 6.3. The Quadratic Formula 6.4. Variations on a Theme 6.4.1. Rearrangements 6.4.2. Substitutions 6.5. Square Roots of Irrationals and Imaginaries 6.6. Beyond Quadratics Problems to Solve for Chapter 6 the BIG PICTURE 7. Special Factorizations and Clever Manipulations 7.1. Factorizations 7.2. Manipulations Problems to Solve for Chapter 7 the BIG PICTURE 8. What Numbers Really Are 8.1. Integers and Rationals 8.2. Lowest Terms and Irrationals 8.3. Complex and Beyond Problems to Solve for Chapter 8 9. An Introduction to Circles the BIG PICTURE 10. Angles 10.1. Lines, Rays, and Segments 10.2. Classification and Measurement 10.3. Angles and Parallel Lines 10.4. Arcs, Segments, Sectors, and Angles 10.5. Angles Formed By Lines Intersecting a Circle 10.6. The Burden of Proof 11. Triangles, a.k.a. Geometry 11.1. Classifying Triangles 11.2. Parts of a Triangle 11.3. The Triangle Inequality 11.4. The Pythagorean Theorem 11.5. Congruent Triangles 11.6. Similar Triangles 11.7. Introduction to Trigonometry 11.8. Area of a Triangle 11.9. A Handful of Helpful Hints Problems to Solve for Chapter 11 the BIG PICTURE 12. Quadrilaterals 12.1. The Fundamentals 12.2. Trapezoids 12.3. Parallelograms 12.4. Rhombuses (Rhombi?) 12.5. Rectangles and Squares 12.6. Hints and Problems Problems to Solve for Chapter 12 13. Polygons 13.1. Types of Polygons 13.2. Angles in a Polygon 13.3. Regular Polygons 13.4. Regular Hexagons Problems to Solve for Chapter 13 14. Angle Chasing Problems to Solve for Chapter 14 15. Areas 15.1. Similar Figures 15.2. Same Base/Same Altitude 15.3. Complicated Figures Problems to Solve for Chapter 15 the BIG PICTURE 16. The Power of Coordinates 16.1. Labelling the Plane 16.2. What's it Good For? 16.3. Straight and Narrow 16.4. Plotting a Line 16.5. The Distance Formula and Circles 16.6. Went Down to the Crossroads . . . 16.7. . . . Fell Down on My Knees Problems to Solve for Chapter 16 17. Power of a Point 17.1. Introduction 17.2. Power of a Point Proofs Problems to Solve for Chapter 17 18. Three Dimensional Geometry 18.1. Planes, Surface Area, and Volume 18.2. Spheres 18.3. Cubes and Boxes 18.4. Prisms and Cylinders 18.5. Pyramids and Cones 18.6. Polyhedra 18.7. How to Solve 3D Problems Problems to Solve for Chapter 18 19. Shifts, Turns, Flips, Stretches, and Squeezes 19.1. Translation 19.2. Rotation 19.3. Reflection 19.4. Distortion 19.5. Dilation 19.6. The More Things Change . . . 19.7. Transformation Proofs Problems to Solve for Chapter 19 20. A Potpourri of Geometry Problems to Solve for Chapter 20 21. Functions 21.1. Welcome to the Machine 21.2. Graphing Functions 21.3. Inputs and Outputs 21.4. Even and Odd 21.5. Some Special Functions 21.5.1. Absolute Values 21.5.2. Floored 21.5.3. Split Up 21.6. Transforming a Function Problems to Solve for Chapter 21 22. Inequalities 22.1. What They Do 22.2. Linear Inequalities 22.3. Quadratic Inequalities 22.4. Absolute Value Inequalities 22.5. A Trivial Inequality Problems to Solve for Chapter 22 the BIG PICTURE 23. Operations and Relations 23.1. What is an Operation? 23.2. Properties of Operations 23.3. Relations Problems to Solve for Chapter 23 the BIG PICTURE 24. Sequences and Series 24.1. Arithmetic Series 24.2. Geometric Series 24.3. Infinite Series 24.4. $\sum^n_{i=1}$ 24.5. Sequences 24.6. Sequences and Means Problems to Solve for Chapter 24 the BIG PICTURE 25. Learning to Count 25.1. What's to Learn? 25.2. Multiplication 25.3. Example: The Number of Divisors 25.4. Restrictions on Multiplication 25.5. Permutations, Arrangements, and ! 25.6. Mixing it Up 25.7. Counting the Wrong Thing, Part I 25.8. Counting the Wrong Thing, Part II 25.9. Doing it Another Way 25.10. The Binomial Theorem Problems to Solve for Chapter 25 26. Statistics and Probability 26.1. Statistics 26.2. Probability and Common Sense 26.3. Multiplying Probabilities 26.4. Casework 26.5. Odds 26.6. What Did You Expect? Problems to Solve for Chapter 26 the BIG PICTURE 27. Sets 27.1. Some Definitions 27.2. Operating on Sets 27.3. Venn Diagrams 27.4. Subsets Problems to Solve for Chapter 27 the BIG PICTURE 28. Prove It 28.1. Words, Words, Words 28.2. Contradiction 28.3. Converses Aren't Necessarily True 28.4. Mathematical Induction 28.5. Shooting Holes in Pigeons 28.6. Convincing But Wrong Problems to Solve for Chapter 28 the BIG PICTURE 29. Parting Shots Problems to Solve for Chapter 29 Index Grades 8 -11. The Art of Problem Solving, Volume 1, is the classic problem solving textbook used by many successful MATHCOUNTS programs, and have been an important building block for students who, like the authors, performed well enough on the American Mathematics Contest series to qualify for the Math Olympiad Summer Program which trains students for the United States International Math Olympiad team. Volume 1 is appropriate for students just beginning in math contests. MATHCOUNTS and novice high school students particularly have found it invaluable. Although the Art of Problem Solving is widely used by students preparing for mathematics competitions, the book is not just a collection of tricks. The emphasis on learning and understanding methods rather than memorizing formulas enables students to solve large classes of problems beyond those presented in the book. Speaking of problems, the Art of Problem Solving, Volume 1, contains over 500 examples and exercises culled from such contests as MATHCOUNTS, the Mandelbrot Competition, the AMC tests, and ARML. Full solutions (not just answers!) are available for all the problems in the solution manual. The Art of Problem Solving, Volume 1, is the classic problem solving textbook used by many successful MATHCOUNTS programs, and have been an important building block for students who, like the authors, performed well enough on the American Mathematics Contest series to qualify for the Math Olympiad Summer Program which trains students for the United States International Math Olympiad team. Volume 1 is appropriate for students just beginning in math contests. MATHCOUNTS and novice high school students particularly have found it invaluable. Although the Art of Problem Solving is widely used by students preparing for mathematics competitions, the book is not just a collection of tricks. The emphasis on learning and understanding methods rather than memorizing formulas enables students to solve large classes of problems beyond those presented in the book. Speaking of problems, the Art of Problem Solving, Volume 1, contains over 500 examples and exercises culled from such contests as MATHCOUNTS, the Mandelbrot Competition, the AMC tests, and ARML. Full solutions (not just answers!) are available for all the problems in the solution manual. - Publisher. "This is the classic problem solving textbook used by many successful Mathcounts programs, and have been an important building block for students who, like the authors, performed well enough on the American Mathematics Contest series to qualify for the Math Olympiad Summer Program which trains students for the United States International Math Olympiad team. Volume 1 is appropriate for students just beginning in math contests. Mathcounts and novice high school students particularly have found it invaluable"-- Publisher description V. 1. The Basics -- V. 1 [pt. 2]. The Basics Solutions Manual -- V. 2. And Beyond -- V. 2 [pt. 2] And Beyond Solutions Manual. Sandor Lehoczky, Richard Rusczyk. Includes Index.