During a certain lecture, each of five mathematicians fell asleep exactly twice. For each pair of mathematicians, there was some moment when both were asleep simultaneously. Prove that, at some moment, three of them were sleeping simultaneously.
Problem
Source:
Tags: AMC, USA(J)MO, USAMO, graph theory, combinatorics unsolved, combinatorics
26.08.2011 13:28
Ages ago, we had this on http://www.artofproblemsolving.com/Forum/viewtopic.php?f=41&t=18809 . I am not sure if this is really pre-olympiad, as the solution requires some olympiad mathematics experience imho. Oh, and I hate to say I still don't understand pbornsztein's solution.
26.08.2011 15:47
It's closely related to this one: https://www.artofproblemsolving.com/Forum/viewtopic.php?f=41&t=422227&p=2387989&hilit=interval#p2387989 It also appeared here (with a solution pointlessly copied out of Kalva): https://www.artofproblemsolving.com/Forum/viewtopic.php?f=41&t=355668&hilit=five+professors and here (with a solution that I haven't looked at yet but maybe which is too simple): https://www.artofproblemsolving.com/Forum/viewtopic.php?f=42&t=82344&hilit=professors+asleep
27.08.2011 02:29
There is also a very nice solution involving graph theory (from ACoPS).