Four vehicles travelled on a road with constant velocities. The car overtook the scooter at 12 o'clock, then met the bike at 14.00 and the motorcycle at 16.00. The motorcycle met the scooter at 17.00 then it overtook the bike at 18.00. At what time did the bike and the scooter meet?

[This is a problem, dating back to 1987, from the Hungarian KöMaL magazine. You'll find many other challenging problems in English on the KöMaL website.]