This was passed to me haphazardly, so I apologize in advance for any mistakes.
You are on a circular lake in a boat. There is a monster at the edge of the lake, waiting to eat you, who can't go in the water. You must reach shore before he reaches you there; then you are safe, regardless of how close he is to you. You can row at speed x, but he can run at speed 4x. What do you do?
NB: assume the monster pursues its optimal strategy, which is to move always to the point on shore to which you are closest.
EDIT: https://puzzling.stackexchange.com/questions/2155/the-lake-monster
Negotiate effectively with the monster. Focus on interests not positions. The monster's position is: I will eat you if you reach the shore and I am there when you get there. Find out what the monster's underlying interest is. Is it hungry? Is it interested in real power, or only in appearing powerful because it has to maintain its monster image and fears ridicule? Maybe the monster simply craves variety and is sick of fish. Does it speak your language? Maybe it's lonely. Get as close to shore as you can but still stay out of reach. Let the monster know that you would like to begin a negotiation and try to uncover the monster's interests.
Just to be clear, this isn't some bullshit where "circles don't actually have edges harharhar" right? My preemptive answer: 1) Choose one direction/point at the edge, and "draw" a line, from you at the center, to that edge. 2) proceed to go in circles relative to the centerpoint, but in a spiral motion so that your completed revolution is at a point slightly further down the line each time. This forces the monster to constantly go in circles while you move infinitely closer to the edge of the circle. 3) You will eventually be on a point infinitely close to the edge of the circle, i.e. the shore, wherein touching the shore doesn't pass any actual time for the monster to be able to be allowed to move to your position. 4)?????? 5) Profit
Turns out I told it correctly: https://puzzling.stackexchange.com/questions/2155/the-lake-monster wasoxygen found it on StackExchange.
Wow, okay, so I didn't even tell the question right but you still got the answer. The gist of it, I think, is that you can row 1 radius and the monster can run 4r and so you need to "shorten the circle" by rowing to r/4, whereupon you can move around the circle faster than the monster can run. I believe. That was roughly my solution (after I sugged 8bit's everlasting spiral) and my friend said I got it, but I may be mixing up the details. I'll try to get back to you with the actual question.
As thundara demonstrated, the problem as stated is quite straightforward: As long as you're within distance r/4 of the center (r being the radius), you can remain on the opposite side of the center from the monster. That means that you have only (3/4)r distance to row straight out, while the monster must run halfway around the lake, a distance of πr. Since πr/4 > (3/4)r, you can simply row straight out from that point and you'll win. In fact, as long as the monster is slower than π + 1≈ 4.14 times as fast as you, you can live without having to make any complicated maneuvers. A more interesting question is: What is the worst case? That is, how fast can the monster run and still be unable to catch you? Seems like we will need curves for this one. Not sure if you need the calculus of variations or if there's a more elementary approach. Update: There's a link on stackexchange to an article at generic maths. This article claims that a straight path is best, although there is a better straight path than the one I gave: if you rowed on a curved line, you'd end up at a location that you could have gotten to quicker with a straight shot.
Seems reasonable, but not entirely convincing. Update 2: I'm convinced now.
galen’s is the best solution. But if it is vetoed, here's an effective strategy: If you are not at the center, row to the center. From the center, identify the point directly across from the monster and row straight to it. Followup: how fast is the fastest monster you can escape this way? And how do you escape it? I suspect that we could escape a very fast monster with adroit rowing.