Given the speed of A, find the speed of B and C.

The circular track has 2*pi radians and 120 meters long so people running round it change angle at a rate of 2*pi/120= pi/60 radians per meter.

If B is running with speed v then he is changing direction at a rate of v*(pi/60) radians per second.

After a time t B will have gone v*t*(pi/60) radians

Putting the circle in the x-y plane B's x co-ordinate is r*cos(v*t*(pi/60)) where r is the radius of the circle

B's y co-ordinate will be r*sin(v*t*(pi/60))

Likewise if A is running at speed u then his x and y co-ordinates will be r*cos(u*t*(pi/60)) and r*sin(u*t*(pi/60))

C is running in the opposite direction at a speed w. Because its going the opposite direction you have to put a minus sign in front, so C's x co-ordinate is -r*cos(w*t*(pi/60)) and y co-ordinate is -r*sin(w*t*(pi/60))

Now you have their co-ordinates in terms of time get an expression for the distance between their points.

Distance |BC|= |AB|

That gives you one equation in terms of time, solve the equation for t.

Remember all angles are in radians.