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We present here the English version of the Romanian first edition (V. Berinde: Ex plorare, investigare si descoperire in matematica, Editura Efemeride, Baia Mare, 2001). There are no major changes. Only a few printing errors were corrected. When transcribing Romanian names or denominations we did not use the diacrit ical marks. Our purpose is to provide an introduction to creative problem solving tech niques with particular emphasis on how to develop inventive skills in students. We present an array of 24 carefully selected themes that range over all the main chapters in elementary mathematics: arithmetic, algebra, geometry, analysis as well as applied mathematics. Main goal is to offer a systematic illustration of how to organize the natural transition from problem solving activity toward exploring, investigating and discovering new facts and results. The book is addressed mainly to students, young mathematicians, and teach ers, involved or/and actively working in mathematics competitions and training gifted people. It collects many valuable techniques for solving various classes of difficult problems and, simultaneously, offers a comprehensive introduction to cre ating new problems. The book should also be of interest to anybody who is in any way connected to mathematics or interested in the creative process and in mathematics as an art.
List of contents
Preface Introduction.- 1 Chase problems.- 2 Sequences of Integers Simultaneously Prime.- 3 A Geometric Construction Using Ruler and Compass.- 4 Solving a Class of Nonlinear Systems.- 5 A Class of Homogenous Inequalities.- 6 The First Decimal of Some Irrational Numbers.- 7 Polynomial Approximation of Continuous Functions.- 8 On an Interesting Divisibility Problem.- 9 Determinants with Alternate Entries.- 10 Solving Some Cyclic Systems.- 11 On a Property of Recurrent Affine Sequences.- 12 Binomial Characterizations of Arithmetic Progressions.- 13 Using Duality in Studying Homographic Recurrences.- 14 Exponential Equations Having Exactly Two Solutions.- 15 A Class of Functional Equations.- 16 An Extension of the Leibniz-Newton Formula.- 17 A Measurement Problem.- 18 A Class of Discontinuous Functions Admitting Primitives.- 19 On Two Classes of Inequalities.- 20 Another Problem of Geometric Construction.- 21 How Can We Discover New Problems by Means of the Computer.- 22 On the Convergence of Some Sequences of Real Numbers.- 23 An Application of the Integral Mean.- 24 Difference and Differential Equations.- Addendum.
Additional text
"In the spirit of George Polya's classic treatise How to Solve It: A New Aspect of Mathematical Method, this book is Vasile Berinde's attempt to "algorithmetize" the creative process involved in problem-solving.... The author's stated goals are twofold. The most obvious aim is to provide methods for solving some difficult problems. But on a deeper level, he is trying to grow new research mathematicians by planting the seeds that will develop into the skills they will need to do original research.... Each of the 24 chapters begins with a source or starting problem and its solution. Then the fun begins! Remarks about the "essence" of the problem and its solution suggest new directions to explore, and these investigations lead to generalizations and the formulation of related problems, sometimes building a "factory" of new problems.
From the first page I eagerly grabbed my pencil and a stack of scratch paper, and set to work as I happily read along.... I recommend the book for all lovers of mathematics, but especially students and teachers who participate in mathematics contests and practice problem solving."
—MAA Online
Report
"In the spirit of George Polya's classic treatise How to Solve It: A New Aspect of Mathematical Method, this book is Vasile Berinde's attempt to "algorithmetize" the creative process involved in problem-solving.... The author's stated goals are twofold. The most obvious aim is to provide methods for solving some difficult problems. But on a deeper level, he is trying to grow new research mathematicians by planting the seeds that will develop into the skills they will need to do original research.... Each of the 24 chapters begins with a source or starting problem and its solution. Then the fun begins! Remarks about the "essence" of the problem and its solution suggest new directions to explore, and these investigations lead to generalizations and the formulation of related problems, sometimes building a "factory" of new problems.
From the first page I eagerly grabbed my pencil and a stack of scratch paper, and set to work as I happily read along.... I recommend the book for all lovers of mathematics, but especially students and teachers who participate in mathematics contests and practice problem solving."
-MAA Online
