The duck in the pond and the fox on the shore
A duck is at the centre of a circular pond of radius R. A fox waits on the shore. The duck can fly away the instant it reaches the shore, but not before. The fox runs along the shore at four times the duck's swimming speed and always positions itself as close to the duck as possible.
Can the duck reach the shore without the fox being there to meet it? If so, how?
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Near the centre, the duck can go round a small circle faster (in angle) than the fox can go round the big one. Use that to get the fox directly opposite. Then compare a straight dash to the shore with the fox's half-lap.
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