Four roads on a plane, each a straight line, are in general position so that no two are parallel and no three pass through the same point. Along each road walks a traveler at a constant speed. Their speeds, however, may not be the same. It's known that two of the travelers have met each of the other three already.
Prove that the other two have also met each other.
The problem can be generalized to an arbitrary number of roads, which makes it even more striking: Assume that two of the travelers met and have each met all the remaining fellows.
Prove that, if this is the case, the remaining ones all have met each other (ie, if two travelers have met everyone, then everyone has met everyone).
You say that no three roads pass through the same point. Do you
mean exactly three or three and greater (i.e. can all four pass through
the same point?)?
Another question for Sam
