I think I have a possible solution to the problem proposed in Danny's latest short thoughts.
It took me about 15 minutes, which I guess puts me somewhere in the nerdiness scale between Eric and Joe/Dave, assuming that it's correct. I'm not totally satisfied with it, but it has some merit.
UPDATE: Finally definitively solved it after some more thinking (another 15 minutes or so). If you don't want it to be spoiled, don't click the "Read more" link below.
Let's think in terms of polar coordinates to make the explanation simpler. Then assume you have a linear velocity of R units/s just to make the math simpler -- it doesn't matter what you have, since the monster's speed is just a constant factor of your speed, ie 4. That means at the bank, your maximum angular velocity is 1 radian/s, and the monster, whose movement is constrained to the bank, has a maximum angular velocity of 4 radians/s.
[The right solution] First, move in the direction diametrically opposite to that of the monster for some time less than (p-3)/4 seconds. Let's say 0.025 seconds to be concrete. This gets the monster moving to reach a point 180 degrees in the opposite direction.
Since the monster moves at angular velocity 4 radians/s, it will have moved 0.1 radians in this time. Now, the angle between you and the monster is (p-0.1) radians.
The monster can choose either to move counterclockwise or clockwise -- it doesn't matter. Now, after 0.025 seconds, you start to move with angular velocity exactly 4 radians second, while also moving radially outward in a spiral in the same direction as the monster, such that you're moving at your maximum linear velocity of R units/sec. This means that just by spiraling, you can never exceed a distance R/4 from the center, since at that point, all your velocity is angular. But we won't reach that stage...
As you can see, the monster must move along the bank in the same direction as you're spiralling to preserve the minimum distance relationship. Because both of you are moving at 4 radians/sec, the relative angle between the two of you will remain constant, (p-0.1) radians.
In math terms:
q'(t) = 4
r'(t) = sqrt(R2 - 16r2)
You can solve the differential equation yourself... I don't feel like writing it out. =P
Anyhow, once you reach a distance less than (p-0.1)R/4 from the circumference of the circle (possible since it's less than R/4 away from the center of the circle), head straight for the circumference. This will allow you to reach the bank before the monster hits, since it must travel an angle of (p-0.1) radians, which will take it more time than you will spend in reaching the edge. Done!
[The original flawed solution]
The idea is to move, with constant linear velocity, in a semicircular path of radius R/2, whose diameter is perpendicular to the line from the center (you) to the monster's present position. Initially, you move in a direction towards the monster, and draw a semicircle so that when you reach the bank, you're moving 180 degrees from your original starting direction.
This should "work" because the monster can't "anticipate" your future movements, it can only react to them. This means that in tracking you, it can move only as fast as your angular velocity. Initially, your angular velocity is 0, increasing to a maximum of 1 radian/sec when you're arriving at the bank. Thus you and the monster "arrive" at the same point on the bank at the "same" time, rather than having it wait for you already there.
The above is not totally mathematically satisfying because technically, you and the monster still reach the same point in the same time, but philosophically speaking, this solution is acceptable to me because in the real world the information about where you're moving has some non-zero propagation time and there would be some reaction time by the monster (since it is "reacting" to your movements as well, not just "standing there" with zero movement). Thus you'd still "reach" the point of bank before the monster does.
Let me think about this some more though...