Les livrets de la collection ATOM (« Aime-T-On les Mathématiques », en anglais : « A Taste of Mathematics ») sont publiés par la Société mathématique du Canada (SMC). Ce sont des ressources d’enrichissement pour les élèves du secondaire qui sont doués en mathématiques et qui s’y intéressent. Certains livrets de la collection aborderont aussi des connaissances utiles en vue d’une participation à des concours mathématiques nationaux et internationaux.

**Publié par :** la Société mathématique du Canada (SMC)**Rédacteur en chef :** Shawn Godin**Conseil de rédaction :** cliquez ici**Rédactrice-gérante :** Termeh Kousha (SMC)

## Livrets dans la série

**Dernier numéro :**

**ATOM Volume XVII : Mathematical Logic Puzzles on a Grid**

*Susan Milner (University of the Fraser Valley)*

This book is intended to introduce secondary students to the joys of logical reasoning by way of puzzles. The four types of puzzles described herein have been successfully shared with people from ages 12 to 90. The hope is that students, teachers, and any curious puzzle enthusiasts will find the collection accessible, enjoyable, and a gateway to increasingly challenging puzzles. No prior mathematical knowledge is assumed aside from basic numeracy.

**Autres numéros :**

*Veuillez noter que les titres dans la série sont répertoriés avec les titres les plus récents en premier.*

*Iliya Bluskov (University of Northern British Columbia)*

Over the years, Iliya Bluskov has collected a huge number of nice problems and solutions on various topics in Mathematics and this publication represents a part of this collection. The publication is written partly as an introductory text, and partly as a book about method of solving, and generally ordered in increasing level of difficulty. The book should be fully accessible to high school students, and parts of it to even younger students and can be used for preparation for mathematical competitions, but it can be useful in preparation for any future work in mathematics. It can be also used by teachers who work on preparation of students for competitions, and by instructors who teach any course that covers recurrence relations.

The problems in this text are from a vast number of sources; ranging from problems proposed by Iliya Bluskov for various competitions or from his class work in relevant courses, problems that were proposed by not used for competitions, or problems from actual competitions such as Olympiads, regional, national and international competitions, and also journal competition problems or proposals ranging from quite obscure and unknown to some classic problems from texts or well-known competition problems.

*Michel Bataille (Rouen, France)*

On constate actuellement un fort déclin de la géométrie dans les programmes de mathématiques de nombreux pays. Dans ces conditions, l’étudiant.e confronté à un problème de géométrie (d’une olympiade, par exemple) peut se sentir à court d’idées bien en peine, par manque de pratique et de connaissances, de découvrir une solution « par la géométrie pure ». La géométrie analytique pourra souvent lui apporter une aide appréciable, en l’emmenant rapidement sur le terrain plus familier de l’algèbre élémentaire. Ce tome de la série ATOM propose de nombreux problèmes, certains classiques, tous traités dans le cadre de la géométrie analytique. Dans les quatres premiers chapitres, après des rappels illustrés d’exemples entièrement traités, plusieurs problèmes sont proposés, tous résolus dans le cinquième chapitre. J’espère ainsi fournir à l’étudiant.e une méthode directe et simple de résolution et par là, renforcer son assurance et aviver son goût pour la géométrie.

Ce livret est en français seulement.

*Margo Kondratieva with Justin Rowsell (Memorial University)*

Secondary school students are often familiar with finite arithmetic and geometric series. Those who attempt a more advanced level of study become introduced to infinite series and some formal techniques of their summation. However, many interesting, non-standard, and important examples remain outside of students’ view and experience.

In this book, while maintaining rigorous approach, we use a more intuitive treatment of the topic. We refer to mostly elementary techniques involving solving algebraic inequalities, linear and quadratic equations. We believe that the ideas we explain and illustrate with many examples can be understood at the secondary school level and help to develop a genuine understanding of the topic. An advanced familiarity with the topic may foster a deeper study of mathematics at the university level.

Some of our problems are connected to Euclidean geometry or reveal other links with topics studied at the secondary school level. We also illustrate how infinite sums may appear while solving some word problems that do not explicitly refer to series and convergence. We talk about some practical applications, such as calculations with an approximation. As well, we introduce some notions and objects that are extremely important in modern mathematics, for example, the Riemann zeta function and the Dirichlet kernel. We hope that reading this book and solving the exercises will stimulate students’ interest and fascination with this amazing area of mathematics.

*Edward J. Barbeau (University of Toronto)*

While the quadratic equation is part of the standard syllabus in secondary school, the scope of this topic has been curtailed in many jurisdictions over the years. It is not enough for students to simply do basic factoring exercises and engage in rote application of the quadratic formula in solving equations. This criticism has even more force when it comes to the topic of complex numbers. For many students, complex numbers arise only in the discussion of the roots of a quadratic equation with negative discriminant. Students have no idea of their theoretical and utilitarian importance in mathematics. This book is intended as a companion to the usual high school fare. The reader is assumed to have been introduced to polynomials and operations of addition, subtraction, multiplication and division, the remainder and factor theorems, simple factorizations, solution of quadratic equations by factoring, completing the square and the quadratic formula, and the relationship between the roots and coefficients of a quadratic equation. In addition, the reader should know the fundamental trigonometric functions and their values at standard angles as well as simple relationships among them.

*Edward J. Barbeau (University of Toronto)*

This book is intended for secondary students with some experience in school geometry. It is assumed that they have had enough elementary Euclidean geometry to cover theorems about congruences of triangles, properties of isosceles and right triangles, basic area theorems for triangles and quadrilaterals, properties of circles and concyclic quadrilaterials. It is expected that the reader would have been introduced to the definitions of translations, rotations and reflections, but has not used them as a tool for solving geometric problems. Many of the solutions are attributed to secondary students who participated in correspondence programs and provided a different perspective on the problems and solutions.

*Bruce L.R. Shawyer & Bruce B. Watson (both of Memorial University)*

The problems in this volume were originally designed for mathematics competitions aimed at students in the junior high school levels (grade 7 to 9) and including those students who may have the talent, ambition and mathematical expertise to represent Canada internationally. The problems herein function as a source of “out of classroom” mathematical enrichment that teachers and parents/guardians of appropriate students may assign to their charges. To aid in this, answers and complete solutions are provided to all the problems (except the relays where there are answers only) and problems and solutions are presented in separate chapters. The authors have also deliberately avoided the temptation to discuss the various mathematical concepts or to intrude in any way with what is done in the school system. This volume is similar to previous publications on *Problems for Mathematics Leagues* in this series.

*Naoki Sato (Art of Problem Solving Inc., San Diego, California)*

We assume that readers are already familiar with basic number theory concepts, such as divisibility and greatest common divisor, but this is not a strict prerequisite – a keen student can begin right away. This volume is to introduce readers to further concepts in number theory, and then show how they can be applied to solve problems. It is meant to serve more as a problem-manual rather than a formal textbook, and proofs of theorems are sometimes replaced by well-chosen examples. And, all solutions are worked out in detail, as they would have to be on a written test. The design of this book is to actually attempt the reader to solve the problems in this booklet, rather than go straight to the solutions. For this purpose, problems are given at the end of each chapter to allow the reader to try to solve them. Many of the problems allow for different approaches, so you may find a solution that is different from ours which is still correct. We hope you enjoy reading and learning about number theory and the problem-solving process and I welcome any suggestions. Happy solving!

*Edward J. Barbeau (University of Toronto)*

This book contains over sixty problems that were originally published in the CAUT Bulletin, published by the Canadian Association of University Teachers. The Canadian Mathematical Society is grateful to the Canadian Association of University Teachers for granting permission for these problems to be included in this ATOM volume. The reader might ask how mathematical problems wound up in a publication whose articles generally deal with matters of university governance, academic policies and conditions of employment of the faculty and librarians of Canada’s colleges and universities.

Most of these problems are not original; the ideas come from a number of sources, some of them school texts. New problems appear on the scene regularly, and a couple of them are included here. I hope that readers enjoy trying their hand at them.

*Peter I. Booth, John F. Grant McLoughlin and Bruce L.R. Shawyer*

This volume is a follow up to our previous publications (Atom 3 and Atom 6) on Problems for Mathematics Leagues. It is the fourth book published by the authors based on their cooperation of devising problems for the Newfoundland and Labrador Senior Mathematics League over a period of more than 16 years. Since the publication of the first ATOM volume, other mathematics leagues, based on our model, have sprung up in other parts of Canada. We are always pleased to assist other leagues, and are prepared to provide current games to help them get started.

*Jim Totten*

Since he first started teaching in 1977, Jim Totten provided his students with a *Problem of the Week*, typically from general interest mathematics but not restricted to it. This generated a healthy collection of problems with 20 problems appearing in each academic year. It is now time to share this collection with others.

The 80 problems in this volume are taken from those posted between 1977 and 1986.

*Peter I. Booth, John F. Grant McLoughlin and Bruce L.R. Shawyer*

This volume is a sequel to Volume 3 and contains more of the problems that have been used in the Newfoundland and Labrador Senior Mathematics League, which is sponsored by the Newfoundland and Labrador Teachers Association Mathematics Special Interest Council. Many of the problems in the booklet admit several approaches. As in Volume 3, this booklet contains no solutions, only answers. Also, the problems are arranged in the form in which we use them — in games. We hope that this will be of use to other groups running Mathematics Competitions.

*Richard Hoshino and John F. Grant McLoughlin*

Mathematical games, puzzles, and contest problems invite a playful spirit of exploration. Quite often, these entertaining and challenging problems are drawn from the fascinating mathematical realm of Combinatorics. This volume is designed to engage those who enjoy playing with mathematics, whether they be teachers, students, or armchair mathematical enthusiasts.

Combinatorial Explorations contains an introduction to Combinatorics through the analysis of three core problems: Handshakes, Routes, and Checkerboards. Each chapter features one of these problems as a springboard for mathematical problem solving. Problem sets, extensions, novel twists, and the inclusion of open-ended investigations offer means through which readers can delve deeper into the mathematics.

*Edward J. Barbeau and Bruce L.R. Shawyer*

This volume contains most of the inequalities that are useful in solving problems. Many inequality problems admit several approaches. Some solutions are given, but other problems are left to the reader.

*Peter I. Booth, John F. Grant McLoughlin and Bruce L.R. Shawyer*

This volume contains a selection of some of the problems that have been used in the Newfoundland and Labrador Senior Mathematics League, which is sponsored the the Newfoundland and Labrador Teachers Association Mathematics Special Interest Council. The support of many teachers and schools is gratefully acknowledged.

We also acknowledge with thanks the assistance from the staff of the Department of Mathematics and Statistics, especially Ros English, Wanda Heath, Menie Kavanagh and Leonce Morrissey, in the preparation of this material.

Many of the problems in the booklet admit several approaches. As opposed to our earlier 1995 book of problems, Shaking Hands in Corner Brook, available from the Waterloo Mathematics Foundation, this booklet contains no solutions, only answers. Also, the problems are arranged in the form in which we use them in games. We hope that this will be of use to other groups running Mathematics Competitions.

*Bruce L.R. Shawyer*

This volume contains a selection of some of the basic algebra that is useful in solving problems at the senior high school level. Many of the problems in the booklet admit several approaches. Some worked examples are shown, but most are left to the ingenuity of the reader.

*Edward J. Barbeau (University of Toronto)*

This volume contains the problems and solutions from the 1995-1996 Mathematical Olympiads’ Correspondence Program. This program has several purposes. It provides students with practice at solving and writing up solutions to Olympiad-level problems, it helps to prepare student for the Canadian Mathematical Olympiad and it is a partial criterion for the selection of the Canadian IMO team.

Thanks are due to Bruce Shawyer of Memorial University of Newfoundland for suggesting the publication of this book and for overseeing its publication, as well as to Cindy Hiscock, 1996 WISE student at Memorial University of Newfoundland, for producing the initial document.

This book is intended to introduce secondary students to the joys of logical reasoning by way of puzzles. The four types of puzzles described herein have been successfully shared with people from ages 12 to 90. The hope is that students, teachers, and any curious puzzle enthusiasts will find the collection accessible, enjoyable, and a gateway to increasingly challenging puzzles. No prior mathematical knowledge is assumed aside from basic numeracy.

Over the years, Iliya Bluskov has collected a huge number of nice problems and solutions on various topics in Mathematics and this publication represents a part of this collection. The publication is written partly as an introductory text, and partly as a book about method of solving, and generally ordered in increasing level of difficulty. The book should be fully accessible to high school students, and parts of it to even younger students and can be used for preparation for mathematical competitions, but it can be useful in preparation for any future work in mathematics. It can be also used by teachers who work on preparation of students for competitions, and by instructors who teach any course that covers recurrence relations.

The problems in this text are from a vast number of sources; ranging from problems proposed by Iliya Bluskov for various competitions or from his class work in relevant courses, problems that were proposed by not used for competitions, or problems from actual competitions such as Olympiads, regional, national and international competitions, and also journal competition problems or proposals ranging from quite obscure and unknown to some classic problems from texts or well-known competition problems.

On constate actuellement un fort déclin de la géométrie dans les programmes de mathématiques de nombreux pays. Dans ces conditions, l’étudiant.e confronté à un problème de géométrie (d’une olympiade, par exemple) peut se sentir à court d’idées bien en peine, par manque de pratique et de connaissances, de découvrir une solution « par la géométrie pure ». La géométrie analytique pourra souvent lui apporter une aide appréciable, en l’emmenant rapidement sur le terrain plus familier de l’algèbre élémentaire. Ce tome de la série ATOM propose de nombreux problèmes, certains classiques, tous traités dans le cadre de la géométrie analytique. Dans les quatres premiers chapitres, après des rappels illustrés d’exemples entièrement traités, plusieurs problèmes sont proposés, tous résolus dans le cinquième chapitre. J’espère ainsi fournir à l’étudiant.e une méthode directe et simple de résolution et par là, renforcer son assurance et aviver son goût pour la géométrie.

Ce livret est en français seulement.

Secondary school students are often familiar with finite arithmetic and geometric series. Those who attempt a more advanced level of study become introduced to infinite series and some formal techniques of their summation. However, many interesting, non-standard, and important examples remain outside of students’ view and experience.

In this book, while maintaining rigorous approach, we use a more intuitive treatment of the topic. We refer to mostly elementary techniques involving solving algebraic inequalities, linear and quadratic equations. We believe that the ideas we explain and illustrate with many examples can be understood at the secondary school level and help to develop a genuine understanding of the topic. An advanced familiarity with the topic may foster a deeper study of mathematics at the university level.

Some of our problems are connected to Euclidean geometry or reveal other links with topics studied at the secondary school level. We also illustrate how infinite sums may appear while solving some word problems that do not explicitly refer to series and convergence. We talk about some practical applications, such as calculations with an approximation. As well, we introduce some notions and objects that are extremely important in modern mathematics, for example, the Riemann zeta function and the Dirichlet kernel. We hope that reading this book and solving the exercises will stimulate students’ interest and fascination with this amazing area of mathematics.

While the quadratic equation is part of the standard syllabus in secondary school, the scope of this topic has been curtailed in many jurisdictions over the years. It is not enough for students to simply do basic factoring exercises and engage in rote application of the quadratic formula in solving equations. This criticism has even more force when it comes to the topic of complex numbers. For many students, complex numbers arise only in the discussion of the roots of a quadratic equation with negative discriminant. Students have no idea of their theoretical and utilitarian importance in mathematics. This book is intended as a companion to the usual high school fare. The reader is assumed to have been introduced to polynomials and operations of addition, subtraction, multiplication and division, the remainder and factor theorems, simple factorizations, solution of quadratic equations by factoring, completing the square and the quadratic formula, and the relationship between the roots and coefficients of a quadratic equation. In addition, the reader should know the fundamental trigonometric functions and their values at standard angles as well as simple relationships among them.

This book is intended for secondary students with some experience in school geometry. It is assumed that they have had enough elementary Euclidean geometry to cover theorems about congruences of triangles, properties of isosceles and right triangles, basic area theorems for triangles and quadrilaterals, properties of circles and concyclic quadrilaterials. It is expected that the reader would have been introduced to the definitions of translations, rotations and reflections, but has not used them as a tool for solving geometric problems. Many of the solutions are attributed to secondary students who participated in correspondence programs and provided a different perspective on the problems and solutions.

The problems in this volume were originally designed for mathematics competitions aimed at students in the junior high school levels (grade 7 to 9) and including those students who may have the talent, ambition and mathematical expertise to represent Canada internationally. The problems herein function as a source of “out of classroom” mathematical enrichment that teachers and parents/guardians of appropriate students may assign to their charges. To aid in this, answers and complete solutions are provided to all the problems (except the relays where there are answers only) and problems and solutions are presented in separate chapters. The authors have also deliberately avoided the temptation to discuss the various mathematical concepts or to intrude in any way with what is done in the school system. This volume is similar to previous publications on *Problems for Mathematics Leagues* in this series.

We assume that readers are already familiar with basic number theory concepts, such as divisibility and greatest common divisor, but this is not a strict prerequisite – a keen student can begin right away. This volume is to introduce readers to further concepts in number theory, and then show how they can be applied to solve problems. It is meant to serve more as a problem-manual rather than a formal textbook, and proofs of theorems are sometimes replaced by well-chosen examples. And, all solutions are worked out in detail, as they would have to be on a written test. The design of this book is to actually attempt the reader to solve the problems in this booklet, rather than go straight to the solutions. For this purpose, problems are given at the end of each chapter to allow the reader to try to solve them. Many of the problems allow for different approaches, so you may find a solution that is different from ours which is still correct. We hope you enjoy reading and learning about number theory and the problem-solving process and I welcome any suggestions. Happy solving!

This book contains over sixty problems that were originally published in the CAUT Bulletin, published by the Canadian Association of University Teachers. The Canadian Mathematical Society is grateful to the Canadian Association of University Teachers for granting permission for these problems to be included in this ATOM volume. The reader might ask how mathematical problems wound up in a publication whose articles generally deal with matters of university governance, academic policies and conditions of employment of the faculty and librarians of Canada’s colleges and universities.

Most of these problems are not original; the ideas come from a number of sources, some of them school texts. New problems appear on the scene regularly, and a couple of them are included here. I hope that readers enjoy trying their hand at them.

This volume is a follow up to our previous publications (Atom 3 and Atom 6) on Problems for Mathematics Leagues. It is the fourth book published by the authors based on their cooperation of devising problems for the Newfoundland and Labrador Senior Mathematics League over a period of more than 16 years. Since the publication of the first ATOM volume, other mathematics leagues, based on our model, have sprung up in other parts of Canada. We are always pleased to assist other leagues, and are prepared to provide current games to help them get started.

Since he first started teaching in 1977, Jim Totten provided his students with a *Problem of the Week*, typically from general interest mathematics but not restricted to it. This generated a healthy collection of problems with 20 problems appearing in each academic year. It is now time to share this collection with others.

The 80 problems in this volume are taken from those posted between 1977 and 1986.

This volume is a sequel to Volume 3 and contains more of the problems that have been used in the Newfoundland and Labrador Senior Mathematics League, which is sponsored by the Newfoundland and Labrador Teachers Association Mathematics Special Interest Council. Many of the problems in the booklet admit several approaches. As in Volume 3, this booklet contains no solutions, only answers. Also, the problems are arranged in the form in which we use them — in games. We hope that this will be of use to other groups running Mathematics Competitions.

Mathematical games, puzzles, and contest problems invite a playful spirit of exploration. Quite often, these entertaining and challenging problems are drawn from the fascinating mathematical realm of Combinatorics. This volume is designed to engage those who enjoy playing with mathematics, whether they be teachers, students, or armchair mathematical enthusiasts.

Combinatorial Explorations contains an introduction to Combinatorics through the analysis of three core problems: Handshakes, Routes, and Checkerboards. Each chapter features one of these problems as a springboard for mathematical problem solving. Problem sets, extensions, novel twists, and the inclusion of open-ended investigations offer means through which readers can delve deeper into the mathematics.

This volume contains most of the inequalities that are useful in solving problems. Many inequality problems admit several approaches. Some solutions are given, but other problems are left to the reader.

This volume contains a selection of some of the problems that have been used in the Newfoundland and Labrador Senior Mathematics League, which is sponsored the the Newfoundland and Labrador Teachers Association Mathematics Special Interest Council. The support of many teachers and schools is gratefully acknowledged.

We also acknowledge with thanks the assistance from the staff of the Department of Mathematics and Statistics, especially Ros English, Wanda Heath, Menie Kavanagh and Leonce Morrissey, in the preparation of this material.

Many of the problems in the booklet admit several approaches. As opposed to our earlier 1995 book of problems, Shaking Hands in Corner Brook, available from the Waterloo Mathematics Foundation, this booklet contains no solutions, only answers. Also, the problems are arranged in the form in which we use them in games. We hope that this will be of use to other groups running Mathematics Competitions.

This volume contains a selection of some of the basic algebra that is useful in solving problems at the senior high school level. Many of the problems in the booklet admit several approaches. Some worked examples are shown, but most are left to the ingenuity of the reader.

This volume contains the problems and solutions from the 1995-1996 Mathematical Olympiads’ Correspondence Program. This program has several purposes. It provides students with practice at solving and writing up solutions to Olympiad-level problems, it helps to prepare student for the Canadian Mathematical Olympiad and it is a partial criterion for the selection of the Canadian IMO team.

Thanks are due to Bruce Shawyer of Memorial University of Newfoundland for suggesting the publication of this book and for overseeing its publication, as well as to Cindy Hiscock, 1996 WISE student at Memorial University of Newfoundland, for producing the initial document.