The Art of Problem Solving — Volume 1: The Basics
Authors: Sandor Lehoczky & Richard Rusczyk
Published by: AoPS Incorporated (7th Edition, 2006)
Scope: 29 chapters covering Algebra, Number Theory, Geometry, Counting, Probability, Functions, Inequalities, Sequences, Sets, and Proof Techniques — the foundational toolkit for mathematical problem solving and competition preparation.
These notes provide chapter-wise comprehensive coverage of every topic, concept, worked example, and exercise from the book. Each chapter page includes theory explanations, key formulas, solved examples, and all end-of-chapter problems. Use this as a self-contained study guide or alongside the original text.
Part I — Algebra & Number Theory
Chapter 1 — Exponents and Logarithms
Integer exponents, fractional exponents, simplifying radicals, rationalizing denominators, and logarithms. Properties of powers, roots, and log rules.
Chapter 2 — Complex Numbers
The imaginary unit $i$, complex number operations (addition, subtraction, multiplication, division), the complex plane, and modulus.
Chapter 3 — Linear Equations
Solving one-variable and two-variable linear equations, systems of equations, substitution and elimination methods, word problems.
Chapter 4 — Proportions
Direct and inverse proportions, manipulating proportions, cross-multiplication, conversion factors, and percent problems.
Chapter 5 — Using the Integers
Divisibility rules, number bases, last digit problems, modular arithmetic, computational tricks, primes, GCD, and LCM.
Chapter 6 — Quadratic Equations
Factoring quadratics, the quadratic formula, discriminant, Vieta’s formulas, substitutions, rearrangements, square roots of irrationals and imaginaries, higher-degree equations.
Chapter 7 — Special Factorizations and Clever Manipulations
Difference of squares, sum/difference of cubes, Sophie Germain identity, Simon’s Favorite Factoring Trick, algebraic manipulations.
Chapter 8 — What Numbers Really Are
Integers, rationals, irrationals, real numbers, complex numbers, lowest terms, proofs of irrationality.
Part II — Geometry
Chapter 9 — An Introduction to Circles
Circle definitions, radius, diameter, chord, arc, circumference, area, inscribed and central angles.
Chapter 10 — Angles
Lines, rays, segments, angle classification and measurement, angles with parallel lines, arcs, sectors, angles formed by lines intersecting a circle.
Chapter 11 — Triangles, a.k.a. Geometry
Triangle classification, medians, altitudes, angle bisectors, triangle inequality, Pythagorean theorem, congruence, similarity, introduction to trigonometry, area formulas.
Chapter 12 — Quadrilaterals
Fundamentals, trapezoids, parallelograms, rhombuses, rectangles, squares — properties, area formulas, and proofs.
Chapter 13 — Polygons
Types of polygons, interior and exterior angle sums, regular polygons, regular hexagons.
Chapter 14 — Angle Chasing
Techniques for finding unknown angles using parallel lines, triangles, circles, and multi-step deductions.
Chapter 15 — Areas
Area of similar figures, same base/same altitude ratios, areas of complicated composite figures.
Chapter 16 — The Power of Coordinates
Coordinate plane, plotting lines, slope-intercept form, distance formula, circle equations, intersections.
Chapter 17 — Power of a Point
Power of a point theorem, intersecting chords, secant-secant, secant-tangent relationships, proofs.
Chapter 18 — Three Dimensional Geometry
Planes, surface area and volume of spheres, cubes, boxes, prisms, cylinders, pyramids, cones, polyhedra.
Chapter 19 — Transformations
Translation, rotation, reflection, distortion, dilation, properties preserved under transformations, transformation proofs.
Chapter 20 — A Potpourri of Geometry
Miscellaneous geometry problems and techniques — a collection of elegant geometric results and challenging problems.
Part III — Functions, Counting & Beyond
Chapter 21 — Functions
Function definition, graphing, domain and range, even and odd functions, absolute value, floor function, piecewise functions, transforming functions.
Chapter 22 — Inequalities
Linear inequalities, quadratic inequalities, absolute value inequalities, the Trivial Inequality, AM-GM inequality.
Chapter 23 — Operations and Relations
Defining custom operations, properties of operations (commutativity, associativity, identity, inverse), relations.
Chapter 24 — Sequences and Series
Arithmetic series, geometric series, infinite series, convergence, sequences, arithmetic and geometric means.
Chapter 25 — Learning to Count
Counting principle (multiplication), permutations, combinations, restrictions, complementary counting, overcounting, the Binomial Theorem.
Chapter 26 — Statistics and Probability
Mean, median, mode, range, basic probability, multiplying probabilities, casework, odds, expected value.
Chapter 27 — Sets
Set definitions, union, intersection, complement, Venn diagrams, subsets, cardinality, inclusion-exclusion.
Chapter 28 — Prove It
Proof terminology, proof by contradiction, converses, mathematical induction, the Pigeonhole Principle, common fallacies.
Chapter 29 — Parting Shots
A collection of challenging mixed problems that tie together concepts from all previous chapters — the ultimate practice set.
Note: These notes are for personal study and reference. For the full experience, including all diagrams and detailed solutions, refer to the original book by Lehoczky & Rusczyk.