The devil has trapped you in his playground.
The devil knows that you can't cross over the burning boundary of his circle, so he allows you to choose a position within the circle before he starts to chase you down. You know that
- You and the devil move at speeds $V$ and $1$ respectively.
- Both move simultaneously and continuously, in any choice of direction.
- Radius of the circle $R=1$.
- The devil leaves an uncrossable burning track along his trajectory:
You're caught by the devil if the distance between you is $0$. The devil will try to catch you as quickly as possible. You know that an angel is en route to save you, so you move to survive for as long as possible.
Question 1: How long can you manage to survive if $V=1$? How should you move?
Question 2: Suppose now you move twice as fast as the devil, i.e. $V=2$. How long can you manage to survive?
Question 3: As your speed $V$ approaches infinity, how long can you manage to survive?
Notice that you can survive for at least $T=2$ by choosing to stay at the opposite side of the devil. On the other hand, you can't survive indefinitely no matter how fast you move, because the devil can carve the disk into patches of exponentially decreasing areas with you inside, shrinking that area to $0$ in finite time.