Is it possible to determine particles' velocities from their crossings on a ring?

Question

2 particles are constrained to move in a ring. Both particles begin moving at $t=0$ from $p=0$, each particle moving in the opposite direction of the other. It is known that they are moving at different constant velocities. Both particles pass each other (without interaction) 4 times before simultaneously returning to $p=0$.

With this information, how can I find out the ratio between the particles' respective velocities?

It seems to me that this would hold true in any case where $x:y$ is the ratio of the particles' velocities and $|x - y| = 5$. In that case, all the following would qualify: $${-2:3},\ {-3:2},\ {1:6},\ {2:7}$$

If furthermore, I know that the order of the positions where they pass is (in radians): $$\frac{4π}{5},\frac{8π}{5},\frac{2π}{5},\frac{6π}{5}$$ then can I definitively extrapolate the ratio?

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Bonus points to anyone who can figure out what gave me the idea for this question in the first place.

Answer 1

The ratio should be 1:4, as the particles are moving in opposite directions (given in question), the way I perceive it is:

Suppose particles A and B with x and 4x velocity in opposite directions. They start at point p. At the time A travels to 1/4th circumference B is at P again. They must have crossed somewhere between p and the point where A is now. Similarly A travelling every quarter of circumference, B completes the round. After 4 rounds of, A and B both are on p.

If velocities are allowed to be negative, then there can be another answer (that also only one.)

I am really sorry if I have misinterpreted the question, but it seems really simple and your thesis was not understandable.