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Until you are more than 1/4 of the radius away from the rock you can swim fast enough so that you can stay 180 degrees away from the tiger. Regardless of which direction the tiger moves around the circle you swim the other way, always keeping 180 degrees away, and moving outward with leftover energy.

Before long you will be 1/4 of the radius away from the center and the tiger will be 180 degrees away. At this point swim straight to the point on the shore furthest from the tiger. You will be able to get there in 3/4 units of time, while it will take the tiger pi/4 =~ 0.7854 units of time.

Thanks to 'Bob' for this one.

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