M and N walk along a circular track. They start at

Solution

M and N start at 5:00 a.m. from the same point in opposite directions, with speeds of 5 and 2 rounds per hour, respectively.  Since they move in opposite directions, their relative speed is (5 + 2 = 7) rounds per hour. They walk for 1.5 hours (from 5:00 to 6:30), covering a combined distance of (7 x 1.5 = 10.5) rounds.  Since they cross each other once per round, the number of crossings is the whole part of 10.5, which is 10.  Hence, they cross each other 10 times before 6:30 a.m.