At 12:00 noon, Anne, Beth and Carmen begin running laps around a circular

track of length three hundred meters, all starting from the same point on the

track. Each jogger maintains a constant speed in one of the two possible directions

for an indefinite period of time. Show that if Anne’s speed is different

from the other two speeds, then at some later time Anne will be at least one

hundred meters from each of the other runners. (Here, distance is measured

along the shorter of the two arcs separating two runners.)