At time 0, Alice and you both move freely at speed 1 on the plane.
As each hour passes, all copies of Alice will subdivide into 2, each moving at half the speed of the parent. So if left alone, at time 1 there will be 2 copies of Alice each moving at speed 1/2, at time 2 there will be 4 copies of Alice each moving at speed 1/4, etc..
If you catch a copy of Alice moving at speed v, you kill it immediately, but your speed will decrease by v permanently.
Alice wishes to multiply without bound. You want to stop her and put a ceiling on the number of copies.
Question: Can you succeed? Does it depend on your distance to Alice at time 0?
Note: All participants in this game are to be viewed as points moving on the plane. You catch a copy if your distance is zero.