Exploring, Investigating and Discovering in Mathematics

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.

"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