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.
Author(s): Sandor Lehoczky; Richard Rusczyk
Series: the Art of Problem Solving
Edition: 7
Publisher: AoPS Incorporated
Year: 2006
Language: English
Pages: 272
Tags: problem solving, Art of Problem Solving, AoPS, mathematics, contest, olympiad
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
