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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.


Source: Albert, John - Department of Mathematics, University of Oklahoma


Collections: Mathematics