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.