Below are some useful links related to mathematical problem solving.
Alexanderson, Gerald L., Leonard F. Klosinski, and Loren C. Larson. The William Lowell Putnam Mathematical Competition Problems and Solutions, 1965-1984. Washington, DC: Mathematical Association of America, 1985. ISBN: 9780883854419.
All Putnam problems for the period 1965-1984, with rather brief solutions (which were originally published in the American Mathematical Monthly).
Barbeau, Edward, Murray S. Klamkin, and W. O. J. Moser. Five Hundred Mathematical Challenges. Washington, DC: Mathematical Association of America, 1997. ISBN: 9780883855195.
Mathematics is at the high school level, but many problems will still be challenging to undergraduates.
Gilbert, George Thomas, Mark Krusemeyer, and Loren C. Larson. Mathematical Plums. (Dolciani Mathematical Expositions, no. 14.) Washington, DC: Mathematical Association of America, 1979. ISBN: 9780883853009.
Gleason, Andrew M., R. E. Greenwood, and L. M. Kelly. The William Lowell Putnam Mathematical Competition Problems and Solutions 1938-1964. Washington, DC: Mathematical Association of America, 1980. ISBN: 9780883854280.
Consists of solutions to all Putnam problems during the period 1938-1964. Very good exposition with lots of motivation, connections with more general areas, etc.
Greitzer, Samuel L. International Mathematical Olympiads, 1959-1977. (New Mathematical Library, no. 27.) Washington, DC: Mathematical Association of America, 1979. ISBN: 9780883856277.
Halmos, Paul R. Problems for Mathematicians, Young and Old. (Dolciani Mathematical Expositions, no. 12.) Washington, DC: Mathematical Association of America, 1991. ISBN: 9780883853207.
I haven't seen this, but it should be quite entertaining.
Honsberger, Ross. Mathematical Morsels. (Dolciani Mathematical Expositions, no. 3.) Washington, DC: Mathematical Association of America, 1979. ISBN: 9780883853030.
Contains 91 problems (with solutions) obtained from various mathematics journals and requiring nothing beyond freshman mathematics to solve.
———. More Mathematical Morsels. (Dolciani Mathematical Expositions, no. 10.) Washington, DC: Mathematical Association of America, 1996. ISBN: 9780883853146.
Similar in format to Mathematical Morsels, with 57 problems and somewhat more discussion of each problem. Most of the problems are taken from the journal Crux Mathematicorum.
———. Mathematical Gems I. (The Dolciani Mathematical Expositions, no. 1.) Washington, DC: Mathematical Association of America, 1974. ISBN: 9780883853016.
———. Mathematical Gems II. (The Dolciani Mathematical Expositions, no. 2.) Washington, DC: Mathematical Association of America, 1976. ISBN: 9780883853023.
———. Mathematical Gems III. (Dolciani Mathematical Expositions, no. 9.) Washington, DC: Published and distributed by the Mathematical Association of America, 1997. ISBN: 9780883853184.
Not really problem books but rather collections of mathematical essays on topics of interest to problem-solvers. However, many interesting problems are discussed.
———. From Erdös to Kiev Problems of Olympiad Caliber. (Dolciani Mathematical Expositions, no. 17.) Washington, DC: Mathematical Association of America, 1997. ISBN: 9780883853245.
Kedlaya, Kiran Sridhara, Bjorn Poonen, and Ravi Vakil. The William Lowell Putnam Mathematical Competition 1985-2000 Problems, Solutions, and Commentary. MAA problem books series. Washington, DC: Mathematical Association of America, 2002. ISBN: 9780883858073.
Similar to the book by Gleason, et. al. - good exposition and motivation.
Klambauer, Gabriel. Problems and Propositions in Analysis. (Lecture Notes in Pure and Applied Mathematics, no. 49.) New York, NY: CRC, 1979. ISBN: 9780824768874.
Several hundred problems and solutions in the four areas (a) arithmetic and combinatorics, (b) inequalities, (c) sequences and series, and (d) real functions. Difficulty ranges from easy to absurd. Includes some famous classical problems which are "well-known" but for which comprehensible complete solutions were impossible to find.
Klamkin, Murray S. USA Mathematical Olympiads, 1972-1985. (New Mathematical Library, no. 33.) Washington, DC: Mathematical Association of America, 1989. ISBN: 9780883856345.
———. International Mathematical Olympiads, 1978-1985 and Forty Supplementary Problems. (New Mathematical Library, no. 31.) Washington, DC: Mathematical Association of America, 1986. ISBN: 9780883856314.
Klee, Victor, and S. Wagon. Old and New Unsolved Problems in Plane Geometry and Number Theory. (Dolciani Mathematical Expositions, no. 11.) Washington, DC: Mathematical Association of America, 1996. ISBN: 9780883853153.
Many easily stated but open problems. Also includes related exercises with solutions.
Konhauser, Joseph D. E., Daniel J. Velleman, and S. Wagon. Which Way Did the Bicycle Go? And Other Intriguing Mathematical Mysteries. (Dolciani Mathematical Expositions, no. 18.) Washington, DC: Mathematical Association of America, 1996. ISBN: 9780883853252.
191 challenging problems with solutions.
Kürschák, József and Hajos, Gyorgy. Hungarian Problem Book, Based on the Eötvös Competitions, Vol. 2: 1906-1928. New York, NY: Random House, 1963.
Larson, Loren C. Problem-Solving Through Problems. Problem books in mathematics. New York, NY: Springer-Verlag, 1983. ISBN: 9780387908038.
Newman, Donald J. A Problem Seminar. Problem books in mathematics. New York, NY: Springer-Verlag, 1982. ISBN: 9780387907659.
A wonderful collection of elegant and ingenious problems, arranged by subject. Each problem comes with a hint and a solution.
Pólya, George, and Gábor Szegö. Problems and Theorems in Analysis. New York, NY: Springer, 2004. ISBN: 9783540636403.
Pólya, George. Problems and Theorems in Analysis 2. Theory of Functions, Zeros, Polynomials, Determinants, Number Theory, Geometry. New York, NY: Springer, 2004. ISBN: 9783540636861.
An English translation of a famous German classic. Develops the equivalent of a graduate level course in classical analysis (real and complex) based on problem solving. While many of the problems are too sophisticated for contests such as the Putnam Exam, there are still a large number of more accessible problems covering material almost impossible to learn otherwise.
Rabinowitz, Stanley. Index to Mathematical Problems, 1980-1984. (Indexes to mathematical problems, v. 1.) Westford, MA: MathPro Press, 1992. ISBN: 9780962640117.
A huge collection of over 5000 problems from the problem columns of dozens of mathematics journals. No solutions.
Shkliarskii, D. O. The USSR Olympiad Problem Book; Selected Problems and Theorems of Elementary Mathematics. A Series of undergraduate books in mathematics. San Francisco, CA: Freeman, 1962.
Vakil, Ravi. A Mathematical Mosaic Patterns & Problem Solving. Burlington, Ontario: Brendan Kelly Pub, 1997. ISBN: 9781895997040.
Winkler, P. Mathematical Puzzles A Connoisseur's Collection. Natick, Mass: AK Peters, 2003. ISBN: 9781568812014.
Highly recommended!