Berkeley Problems in Mathematics (Problem Books in Mathematics) - Tapa dura

From the reviews of the third edition:

"This new edition has been updated with the most recent exams ... . There are numerous new problems and solutions which were not included in previous editions. It is an invaluable source of problems and solutions for every mathematics student who plans to enter a Ph. D program. ... this book will develop problem-solving skills in areas such as real analysis, multivariable calculus, differential equations, metric spaces, complex analysis, algebra, and linear algebra. ... Tags with the exact exam year provide the opportunity to rehearse complete examinations. ... This new edition has been updated with the most recent exams ... ." (Zentralblatt für Didaktik der Mathematik, November 2004)

"The Mathematics department of the University of California, Berkeley, has set a written preliminary examination to determine whether first year Ph.D. students have mastered enough basic mathematics to succeed in the doctoral program. Berkeley Problems in Mathematics is a compilation of all the ... questions, together with worked solutions ... . All the solutions I looked at are complete ... . Some of the solutions are very elegant. ... This is an impressive piece of work and a welcome addition to any mathematician’s bookshelf." (Chris Good, The Mathematical Gazette, 90:518, 2006)

"During the last twenty-five years problems from written preliminary examinations that are required for the Ph.D. degree at the Mathematics Department of the University of California, Berkeley, have been assembled. ... The book is suited for students in mathematics, physics or engineering. Solutions are well explained, making the book valuable for self-study. The problems have a satisfactory high level, so the book is a rich resource of examples for lecturers as well, who need exercises ... . This book certainly is to be recommended." (Paula Bruggen, Bulletin of the Belgian Mathematical Society, 12:4, 2005)

This book collects approximately nine hundred problems that have appeared on the preliminary exams in Berkeley over the last twenty years. It is an invaluable source of problems and solutions. Readers who work through this book will develop problem solving skills in such areas as real analysis, multivariable calculus, differential equations, metric spaces, complex analysis, algebra, and linear algebra.