超越普特南试题：大学数学竞赛中的方法与技巧【Putnam and Beyond,英文版】

R˘azvan Gelca,Titu Andreescu



Preface
Aproblem book at the college level. Astudy guide for the Putnam competition. Abridge
between high school problem solving and mathematical research. Afriendly introduction
to fundamental concepts and results. All these desires gave life to the pages that follow.
TheWilliam Lowell Putnam Mathematical Competition is the most prestigious mathematics
competition at the undergraduate level in the world. Historically, this annual
event began in 1938, following a suggestion of William Lowell Putnam, who realized
the merits of an intellectual intercollegiate competition. Nowadays, over 2500 students
from more than 300 colleges and universities in the United States and Canada take part
in it. The name Putnam has become synonymous with excellence in undergraduate
mathematics.
Using the Putnam competition as a symbol, we lay the foundations of higher mathematics
from a unitary, problem-based perspective. As such, Putnam and Beyond is a
journey through the world of college mathematics, providing a link between the stimulating
problems of the high school years and the demanding problems of scientific
investigation. It gives motivated students a chance to learn concepts and acquire strategies,
hone their skills and test their knowledge, seek connections, and discover real world
applications. Its ultimate goal is to build the appropriate background for graduate studies,
whether in mathematics or applied sciences.
Our point of view is that in mathematics it is more important to understand why than
to know how. Because of this we insist on proofs and reasoning. After all, mathematics
means, as the Romanian mathematician Grigore Moisil once said, “correct reasoning.’’
The ways of mathematical thinking are universal in today’s science.
Putnam and Beyond targets primarily Putnam training sessions, problem-solving
seminars, and math clubs at the college level, filling a gap in the undergraduate curriculum.
But it does more than that. Written in the structured manner of a textbook, but with
strong emphasis on problems and individual work, it covers what we think are the most
important topics and techniques in undergraduate mathematics, brought together within
the confines of a single book in order to strengthen one’s belief in the unitary nature of
for the reader to solve. And since our problems are true brainteasers, complete solutions
are given in the second part of the book. Considerable care has been taken in selecting the
most elegant solutions and writing them so as to stir imagination and stimulate research.
We always “judged mathematical proofs,’’ as AndrewWiles once said, “by their beauty.’’
Putnam and Beyond is the fruit of work of the first author as coach of the University
of Michigan and Texas Tech University Putnam teams and of the International Mathematical
Olympiad teams of the United States and India, as well as the product of the vast
experience of the second author as head coach of the United States International Mathematical
Olympiad team, coach of the Romanian International Mathematical Olympiad
team, director of the American Mathematics Competitions, and member of the Question
Writing Committee of theWilliam Lowell Putnam Mathematical Competition.
In conclusion, we would like to thank Elgin Johnston, Dorin Andrica, Chris Jeuell,
Ioan Cucurezeanu, Marian Deaconescu, Gabriel Dospinescu, Ravi Vakil, Vinod Grover,
V.V. Acharya, B.J. Venkatachala, C.R. Pranesachar, Bryant Heath, and the students of
the International Mathematical Olympiad training programs of the United States and
India for their suggestions and contributions. Most of all, we are deeply grateful to
Richard Stong, David Kramer, and Paul Stanford for carefully reading the manuscript and
considerably improving its quality. We would be delighted to receive further suggestions
and corrections; these can be sent to rgelca@gmail.com.
May 2007 R˘azvan Gelca
Texas Tech University
Titu Andreescu
University of Texas at Dallas