Problem A1 (87): Find the least number A such that for any two squares of combined area 1, a rectangle of area A exists such that the two squares can be packed into that rectangle (without the interiors of the squares overlapping). You may assume that the sides of the squares will be parallel to the sides of the rectangle.
Problem A2 (6): Let C1 and C2 be circles whose centers are 10 units apart and whose radii are 1 and 3. Find, with proof, the locus of all points M for which there exist points X on C1 and Y on C2 such that M is the midpoint of the line segment XY.
Problem B1 (113): Define a selfish set to be a set which has its own cardinality (number of elements) as an element. Find, with proof, the number of subsets of {1,2,...,n} which are minimal selfish sets, that is, selfish sets none of whose proper subsets is selfish.
Problem B3 (20): Given that {x1,x2,...,xn}={1,2,...,n}, find, with proof, the largest possible value, as a function of n (with n at least 2), of x1*x2 + x2*x3 +...+ x{n-1}*xn + xn*x1.
Stumped? Then see the solutions compiled by David Rusin of Northern Illinois University.
The Problem Archive of American Mathematics Competitions contains problems and solutions from recent Putnam competitions, the US Mathematical Olympiads, and the International Mathematical Olympiads, as well as links to other problem collections. (Some of the problem sets here are in TeX format and may be a bit hard to read for someone unfamiliar with this language. Others are available as postscript files.)
David Rusin has compiled Putnam solutions from postings to the sci.math newsgroup.
G. Alexanderson et al., The William Lowell Putnam Mathematical Competition problems and solutions: 1965 - 1984, Mathematical Association of America, Washington, D.C., 1985.
A.M. Gleason et al., The William Lowell Putnam Mathematical Competition problems and solutions: 1938 - 1964, Mathematical Association of America, Washington, D.C., 1980.
Putnam problems and solutions are also published in the American Mathematical Monthly, the official journal of the Mathematial Association of America, usually in the October issues.