# Confused by a Leningrad Mathematical Olympiad question

The following problem appeared in a Leningrad Mathematical Olympiad:

The map of a subway system is a convex polygon in which no three diagonals are concurrent. There is a station at each vertex and at every intersection of two diagonals. Train runs along entire diagonals, but not necessarily every diagonal. If each station lies on the route of at least one train, prove that it is possible to go from any station to any other station, changing trains at most twice.

However if the map of the subway system is as follows, I can’t see how to go from station A to station B, changing trains at most twice.

My question is: Have I misunderstood the Olympiad question?

I want to try to solve this Olympiad problem once I properly understand it. So please don’t give hints/solution to this Olympiad problem. Thank you.

• There is a station at every intersection of two diagonals. (This is not limited to the diagonals that trains run on.) Your map is missing many of these stations. Jan 23 at 5:35
• @DanielMathias In my diagram, I only drew diagonals that trains run along. I can see how there would be more stations that I didn’t draw and that would allow a passenger to have more transfer choices. But the passenger is still limited to the train routes I have drawn. Jan 23 at 5:53
• "Each station lies on the route of at least one train." The routes you have drawn do not service all stations. As such, your map is not valid for the question. Jan 23 at 5:58
• You have to consider all diagonals. All their pairwise intersections are stations and each station must be serviced meaning that of every pair of diagonals that do intersect (inside the polygon) at least one must be active. Jan 23 at 7:00
• @DanielMathias Thank you for helping me understand the question. Jan 23 at 7:08