The best preparation for the Putnam competition is going over the
problems from previous years. Problems (with solutions) can be
found on Kiran Kedlaya's archive.
There are also three published books, containing all past problems up to
2000. In the books the problems are discussed in detail, and sometimes
given multiple solutions:
Other recommended books are:
The William Lowell Putnam Mathematical Competition: Problems and
Solutions: 1938-1964, by A.M.Gleason, R.E.Greenwood, L.M.Kelly, MAA
- The William Lowell Putnam Mathematical Competition : Problems and
1965-1984 (MAA Problem Book Series) by Gerald L. Alexanderson, MAA
- The William Lowell Putnam Mathematical Competition 1985-2000:
Solutions, and Commentary (MAA Problem Book Series)
by Kiran S. Kedlaya, Bjorn Poonen, Ravi Vakil, MAA (2002).
In addition, various universitites maintain web sites with more resources
for Putnam preparation, including practice problems. See for example the
sites at Stanford, Northwestern, and Wesleyan.
- Problem solving through problems, by Loren Larson. Springer (1992).
- Putnam and beyond, by Titu Andreescu and Razvan Gelca.
Springer (2007), available
- A problem seminar, by Donald J. Newman. Springer (1982).
- Problems from the book, by Titu Andreescu and Gabriel
XYZ Press (2010).
- Problems in real analysis:
advanced calculus on the real axis, by Teodora-Liliana Radulescu,
Vicentiu D. Radulescu and Titu Andreescu. Springer (2009).
- Contests in higher mathematics. Miklos Schweitzer Competitions 1962--1991. Edited by Gabor J. Szekely. Problem Books in Mathematics. Springer-Verlag, New York (1996).
- Mathematical morsels by Ross Honsberger. The Mathematical Association of America (1978).
- The USSR Olympiad problem book by D. O. Shklarsky, N. N. Chentzov and I. M. Yaglom. W. H. Freeman and Company, San Francisco and London (1962).
- Generatingfunctionology, by Herbert S. Wilf,
available online. A. K. Peters (2006).
- Problem-Solving Strategies, by Arthur
Engel. Springer (1998).
- 102 Combinatorial Problems, by Titu Andreescu, Zuming Feng. Birkhauser, Boston (2002).
- 104 Number Theory Problems: From the Training of the USA IMO
by Titu Andreescu et al. Birkhauser, Boston (2006).