A duck is swimming at the center of a circular lake. A fox is waiting at the shore, not able to swim, willing to eat the duck. It may move around the whole lake with a speed four times faster than the duck can swim. As soon as duck reaches the surface, it can fly, but not within the pond. Can the duck always reach the shore without being eaten by the fox? Hint
Fox is chasing duck brainlessly. If duck's angular velocity is (somehow) more than fox, duck may actually create some sort of phase lag.
At radius of slightly less than r/4 the duck can swim in circles, forcing the fox to run around. once the duck is at a phase of pi from fox it starts swimming towards the shore and flies.