Run away from lions in a cage I came across an interesting problem: 

There is a round cage and you are in it. Also two lions are in this cage
  too. The start position is that the distance between you and both lions is
  the diameter of the circle (you are on opposite sides of the cage).
  The speed of the lion is 1 m/s.

And the question is: 

What is the minimal speed you need to have to always run away from lions and never be caught.

Probably it is a bit simpler to search the maximal speed when the lions will catch you. And the result of the original task will be the upper limit of that value. 
I think the radius of the cage doesn't matter - it is only a scale problem. The only important thing is that the cage is round.
There is a similar problem here for one lion. But the answer links to buy some book and I couldn't find where to download it for free =).
And also I wonder if there is a solution for the generalized task with $N$ lions. That looks too complicated but I think the idea is the same - the lions should build a line when you can't run between any two of them and two lions on the ends of a chain will behave like the ones in the two-lions problem. 

First, i don't get why two lions are there. But supose they will move always directly to you. If you would move same speed as them around the cage, they would be coming closer and closer to you until they would finnaly cacht you. Or not? I thing they would be still-closing but never able to catch you, since it will end up that they will move 0.0000001mm just behind you. So the conclusion is that you need to move any larger speed than theirs.
