Putnam Seminar Sept. 17, 2010 Summary: Putnam Seminar Sept. 17, 2010 Here are four problems from the 1996 Putnam competition. A1. 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. A2. 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 . B1. 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. B3. 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 x1x2 + x2x3 + · · · + xn-1xn + xnx1. 1 Collections: Mathematics