A dog hiding in the tall grass at the edge of a circular field spots a fox at the center of the eld. The fox, unable to see the dog but sensing the danger, begins to run at a constant speed along a straight line path towards the safety of a uniformly selected random point at the edge of the eld. At that same instant the dog jumps

the fence and runs straight at the fox at a constant speed M times that of the fox.

Find the probability that the fox will escape the field, and the value of M at which the dog must run to have a 50% chance of catching the fox.

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