 
Summary: Putnam Seminar 2004
More problems from "Mathematical Miniatures".
1. At n distinct points of a circular race course, n cars are ready to start. Each of
them covers the course in an hour. At a given signal every car selects one of the two
possible directions and starts immediately. Whenever two cars meet, both of them
change directions and go on without loss of speed. Show that at a certain moment
each car will be at its starting point.
2. (from the 1973 Moscow Olympiad) Twelve painters live in 12 red and blue houses
built around a circular lane. Each month one of them goes counterclockwise along the
lane and, starting from his own house, repaints the houses according to the following
rule: If a house is red, he paints it blue and passes to the next house; if a house is
blue, he paints it red and goes home. Each painter does this once a year. Prove that
if at least one of the houses is red, then a year later each house will have its initial
color.
3. (from the 1995 USAMO) Let p be an odd prime. The sequence (an)n0 is defined as
follows: a0 = 0, a1 = 1, . . . , ap2 = p2 and, for all n p1, an is the least integer
greater than an1 that does not form an arithmetic progression of length p with any
of the preceding terms. Prove that, for all n, an is the number obtained by writing n
in base p  1 and reading it in base p.
