Players A, B, and C are standing on a circle equidistant from each other...
A is walking in the opposite direction of B, and B is walking in the same direction as C.
The ratio of the speeds of the players A:B:C is 1:2:3. How much distance will player A have traveled when players A, B, C meet simultaneously for the 3rd time, if it is possible? The answer is supposed to be in terms of X where X is the starting distance between any two players.
I received this problem in an interview. I still do not know how to approach it. I was thinking to use polar coordinates (representing each player's position as a linear combination of complex exponentials), but I didn't have enough time to recall all of the formulas.
Let's start by assuming that A, B, and C are situated at the 12:00, 8:00, and 4:00 positions, respectively, and that A is traveling clockwise while B and C are traveling counterclockwise. Clearly, B and C will only intersect at the 4:00 position, and only when C has traveled $(9k-3)X$ for some positive integer. But in that time A will have traveled $(3k-1)X$, landing at the 8:00 position. So they will never simultaneously meet. Other orientations of the initial assumptions lead to the same conclusion.