Building bridges between classical results and contemporary nonstandard problems, this highly relevant work embraces important topics in analysis and algebra from a problem-solving perspective. The book is structured to assist the reader in formulating and proving conjectures, as well as devising solutions to important mathematical problems by making connections between various concepts and ideas from different areas of mathematics. Instructors and motivated mathematics students from high school juniors to college seniors will find the work a useful resource in calculus, linear and abstract algebra, analysis and differential equations. Students with an interest in mathematics competitions must have this book in their personal libraries.
Titu Andreescu received his BA, MS, and PhD from the West University of Timisoara, Romania. The topic of his doctoral dissertation was "Research on Diophantine Analysis and Applications." Professor Andreescu currently teaches at the University of Texas at Dallas. Titu is past chairman of the USA Mathematical Olympiad, served as director of the MAA American Mathematics Competitions (19982003), coach of the USA International Mathematical Olympiad Team (IMO) for 10 years (19932002), director of the Mathematical Olympiad Summer Program (19952002), and leader of the USA IMO Team (19952002). In 2002 Titu was elected member of the IMO Advisory Board, the governing body of the world's most prestigious mathematics competition. Titu is also co-founder and director of the AwesomeMath Summer Program (AMSP). Titu received the Edyth May Sliffe Award for Distinguished High School Mathematics Teaching from the MAA in 1994 and a "Certificate of Appreciation" from the president of the MAA in 1995 for his outstanding service as coach of the Mathematical Olympiad Summer Program in preparing the US team for its perfect performance in Hong Kong at the 1994 IMO. Titus contributions to numerous textbooks and problem books are recognized worldwide.
Dorin Andrica received his PhD in 1992 from Babes-Bolyai University in Cluj-Napoca, Romania, with a thesis on critical points and applications to the geometry of differentiable submanifolds. Professor Andrica has been chairman of the Department of Geometry at Babes-Bolyai" since 1995. Dorin has written and contributed to numerous mathematics textbooks, problem books, articles and scientific papers at various levels. Dorin is an invited lecturer atuniversity conferences around the worldAustria, Bulgaria, Czech Republic, Egypt, France, Germany, Greece, Italy, the Netherlands, Portugal, Serbia, Turkey, and the USA. He is a member of the Romanian Committee for the Mathematics Olympiad and member of editorial boards of several international journals. Also, he is well known for his conjecture about consecutive primes called "Andrica's Conjecture." Dorin has been a regular faculty member at the CanadaUSA Mathcamps between 2001-2005 and at the AwesomeMath Summer Program (AMSP) since 2006.
Zuming Feng graduated with a PhD from Johns Hopkins University with emphasis on Algebraic Number Theory and Elliptic Curves. He teaches at Phillips Exeter Academy. Zuming also served as a coach of the USA IMO team (1997-2006), was the deputy leader of the USA IMO Team (2000-2002), and an assistant director of the USA Mathematical Olympiad Summer Program (1999-2002). He is a member of the USA Mathematical Olympiad Committee since 1999, and has been the leader of the USA IMO team and the academic director of the USA Mathematical Olympiad Summer Program since 2003. Zuming is also co-founder and academic director of the AwesomeMath Summer Program (AMSP). Zuming received the Edyth May Sliffe Award for Distinguished High School Mathematics Teaching from the MAA in 1996 and 2002.
Preface.- Glossary of Notation.- Cardinality.- Density.- Lemma of the Closed Intervals.- Sequences Given by Implicit Relations.- Recurrence Relations.- Complementary Sequences.- Quadratic Functions, Quadratic Equations.- Polynomial Functions Involving Determinants.- A Decomposition Theorem Related to the Rank of a Matrix.- Intermediate Value Property.- Uniform Continuity.- Toeplitz Theorem.- Derivatives and Functions Variation.- Weierstrass Theorem.- The Number e.- Riemann Sums, Darboux Sums.- References.- Subject Index.