Discussions of these problems have already started on AoPS. Let's start a few over here. :)
For each integer , define the sequence for as
Determine all values of such that there exists a number such that for infinitely many values of .
Let be the set of real numbers. Determine all functions such that, for any real numbers and ,
A hunter and an invisible rabbit play a game in the Euclidean plane. The rabbit's starting point, and the hunter's starting point, are the same. After rounds of the game, the rabbit is at point and the hunter is at point In the round of the game, three things occur in order:
The rabbit moves invisibly to a point such that the distance between and is exactly .
A tracking device reports a point to the hunter. The only guarantee provided by the tracking device to the hunter is that the distance between and is at most .
The hunter moves visibly to a point such that the distance between and is exactly .
Is it always possible, no matter how the rabbit moves, and no matter what points are reported by the tracking device, for the hunter to choose her moves so that after rounds, she can ensure that the distance between her and the rabbit is at most ?