This is a challenging problem-solving book in Euclidean geometry, assuming nothing of the reader other than a good deal of courage. Topics covered include cyclic quadrilaterals, power of a point, homothety, and triangle centers. Along the way the reader will meet such classical gems as the nine-point circle, the Simson line, the symmedian, and the mixtilinear incircle, as well as the theorems of Euler, Ceva, Menelaus, and Pascal. Another part is dedicated to the use of complex numbers and barycentric coordinates, granting the reader both a traditional and computational viewpoint of the material. The final part consists of some more advanced topics, such as inversion in the plane, the cross ratio and projective transformations, and the theory of the complete quadrilateral. The exposition is friendly and relaxed, and accompanied by over 248 beautifully drawn figures. The emphasis of this book is placed squarely on the problems. Each chapter contains carefully chosen worked examples, which explain not only the solutions to the problems but also describe in close detail how one would invent the solution to begin with. The text contains a selection of 300 practice problems of varying difficulty from contests around the world, with extensive hints and selected solutions. This book is especially suitable for students preparing for national or international mathematical olympiads, or for teachers looking for a text for an honors class.

This book is a celebration of mathematical problem solving at the level of the high school American Invitational Mathematics Examination. There is no other book on the market focused on the AIME. It is intended, in part, as a resource for comprehensive study and practice for the AIME competition for students, teachers, and mentors. After all, serious AIME contenders and competitors should seek a lot of practice in order to succeed. However, this book is also intended for anyone who enjoys solving problems as a recreational pursuit. The AIME contains many problems that have the power to foster enthusiasm for mathematics – the problems are fun, engaging, and addictive. The problems found within these pages can be used by teachers who wish to challenge their students, and they can be used to foster a community of lovers of mathematical problem solving! There are more than 250 fully-solved problems in the book, containing examples from AIME competitions of the 1980's, 1990's, 2000's, and 2010's. In some cases, multiple solutions are presented to highlight variable approaches. To help problem-solvers with the exercises, the author provides two levels of hints to each exercise in the book, one to help stuck starters get an idea how to begin, and another to provide more guidance in navigating an approach to the solution. Topics include algebra, combinatorics, probability, number theory, sequences and series, logarithms, trigonometry, complex numbers, polynomials, and geometry. Many AIME problems involve using several of these mathematical disciplines simultaneously. To make the book indeed "gentle", the author has prepared well-motivated, detailed, clear solutions to all examples and exercises in the book. He has not always opted for the shortest solution, but rather, he has presented solutions that a typical student might say "I could have come up with that!" The book avoids excessive reliance on heavy machinery, deep theorems, or undue abstraction.

This guide covers the story of trigonometry. It is a swift overview, but it is complete in the context of the content discussed in beginning and advanced high-school courses. The purpose of these notes is to supplement and put into perspective the material of any course on the subject you may have taken or are currently taking. (These notes will be tough going for those encountering trigonometry for the very first time!)

A Friendly Mathematics Competition tells the story of the Indiana College Mathematics Competition (ICMC) by presenting the problems, solutions, and results of the first 35 years of the ICMC. The ICMC was organized in reaction to the Putnam Exam—its problems were to be more representative of the undergraduate curriculum, and students could work on them in teams. Originally participation was restricted to the small, private colleges and universities of the state, but was later opened up to students from all of the schools in Indiana. The competition was quickly nicknamed the 'Friendly' Competition because of its focus on solving mathematical problems, which brought faculty and students together, rather than on the competitive nature of winning. Organized by year, the problems and solutions in this volume present an excellent archive of information about what has been expected of an undergraduate mathematics major over the past 35 years. With more than 245 problems and solutions, the book is also a must buy for faculty and students interested in problem-solving. The index of problems lists problems in: Algebraic Structures; Analytic Geometry, Arclength, Binomial Coefficients, Derangements, Differentiation, Differential Equations, Diophantine Equations, Enumeration, Field and Ring Theory, Fibonacci Sequences, Finite Sums, Fundamental Theorem of Calculus Geometry, Group Theory, Inequalities, Infinite Series, Integration, Limit Evaluation, Logic, Matrix Algebra, Maxima and Minima Problems, Multivariable Calculus, Number Theory, Permutations, Probability, Polar Coordinates, Polynomials, Real Valued Functions Riemann Sums, Sequences, Systems of Equations, Statistics, Synthetic Geometry, Taylor Series, Trigonometry, and Volumes.

The Eötvös Mathematics Competition is the oldest high school mathematics competition in the world, dating back to 1894. This book is a continuation of Hungarian Problem Book III and takes the contest through 1963. Forty-eight problems in all are presented in this volume. Problems are classified under combinatorics, graph theory, number theory, divisibility, sums and differences, algebra, geometry, tangent lines and circles, geometric inequalities, combinatorial geometry, trigonometry and solid geometry. Multiple solutions to the problems are presented along with background material. There is a substantial chapter entitled "Looking Back," which provides additional insights into the problems. Hungarian Problem Book IV is intended for beginners, although the experienced student will find much here. Beginners are encouraged to work the problems in each section, and then to compare their results against the solutions presented in the book. They will find ample material in each section to help them improve their problem-solving techniques.

Every mathematician (beginner, amateur, and professional alike) thrills to find simple, elegant solutions to seemingly difficult problems. Such happy resolutions are called 'aha! solutions,' a phrase popularized by mathematics and science writer Martin Gardner. Aha! solutions are surprising, stunning, and scintillating: they reveal the beauty of mathematics. This book is a collection of problems with aha! solutions. The problems are at the level of the college mathematics student, but there should be something of interest for the high school student, the teacher of mathematics, the 'math fan,' and anyone else who loves mathematical challenges. This collection includes 100 problems in the areas of arithmetic, geometry, algebra, calculus, probability, number theory, and combinatorics. The problems start out easy and generally get more difficult as you progress through the book. A few solutions require the use of a computer. An important feature of the book is the bonus discussion of related mathematics that follows the solution of each problem. This material is there to entertain and inform you or point you to new questions. If you don’t remember a mathematical definition or concept, there is a Toolkit in the back of the book that will help.

In the year 2000, the Mathematical Association of America initiated the American Mathematics Competitions 10 (AMC 10) for students up to grade 10. The Contest Problem Book VIII is the first collection of problems from that competition covering the years 2001-2007. J. Douglas Faires and David Wells were the joint directors of the AMC 10 and AMC 12 during that period, and have assembled this book of problems and solutions. There are 350 problems from the first 14 contests included in this collection. A Problem Index at the back of the book classifies the problems into the following major subject areas: Algebra and Arithmetic, Sequences and Series, Triangle Geometry, Circle Geometry, Quadrilateral Geometry, Polygon Geometry, Counting Coordinate Geometry, Solid Geometry, Discrete Probability, Statistics, Number Theory, and Logic. The major subject areas are then broken down into subcategories for ease of reference. The Problems are cross-referenced when they represent several subject areas.

This is the ninth book of problems and solutions from the American Mathematics Competitions (AMC) contests. It chronicles 325 problems from the 13 AMC 12 contests given in the years 2001 through 2007. The authors were the joint directors of the AMC 12 and AMC 10 competitions during that period. The problems have all been edited to ensure that they conform to the current style of the AMC 12 competitions, the graphs and figures have been redrawn to make them more consistent in form and style, and the solutions to the problems have been both edited and supplemented. A Problem Index at the back of the book classifies the problems into the subject areas of Algebra, Arithmetic, Complex Numbers, Counting Functions, Geometry, Graphs, Logarithms, Logic, Number Theory, Polynomials, Probability, Sequences, Statistics, and Trigonometry. A problem that uses a combination of these areas is listed multiple times.

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. First Steps for Math Olympians 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. First Steps for Math Olympians 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.

Contest Problem Book VII chronicles 275 problems from the American Mathematics Contests (AMC 12 and AMC 10 for the years 1995 through 2000, including the 50th Anniversary AHSME issued in 1999. Twenty-three additional problems with solutions are included. A Problem Index classifies the 275 problems in to the following subject areas: Algebra, Complex Numbers, Discrete Mathematics (including Counting Problems), Logic, and Discrete Probability, Geometry (including Three Dimensional Geometry), Number Theory (including Divisibility, Representation, and Modular Arithmetic), Statistics, and Trigonometry. For over 50 years many excellent exams have been prepared by individuals throughout our mathematical community in the hope that all secondary school students will have an opportunity to participate in these problem solving and enriching mathematics experiences. The American Mathematics Contests are intended for everyone from the average student at a typical school who enjoys mathematics to the very best student at the most special school.

It seems to me that anyone who is interested in mathematics and is exposed to high school age kids, whether they are teachers themselves, parents, neighbors, or even as university professors getting involved in our communities through the high schools, would do well to read this book to see some excellent examples of how we can present real mathematics in ways that students can understand. – Darren Glass, MAA Online Reviews The International Mathematical Olympiad (IMO) is a widely renowned international competition in high school mathematics that has been conducted annually since 1959 (with the single exception of the year 1980). Several hundred incredibly gifted high school students from several dozen countries compete each year, taking a two-day examination consisting of three truly difficult problems to be solved each day. The IMO is a challenge to the mental skills of its participants. There are those who win medals and those who, while losing in the narrow sense, gain immeasurably from the intensity of the effort and the competition. This book is a sequel to two earlier MAA publications (NML27 and NML31) by Samuel L. Greitzer and Murray S. Klamkin, which covered the first twenty-six International Mathematical Olympiads. This volume includes all of the Olympiad problems from 1986 through 1999. Each problem is given with a solution (and often more than one). Occasionally, solutions are accompanied by remarks, in which alternative approaches, generalizations of the problem, or the relevance of the problem to some mathematical theory are given. The book is addressed mainly to high school students preparing for participating in mathematics competitions, and to their teachers. However, anyone interested in problem solving will enjoy these problems.