The Lion and the Man Puzzle - Can the Man Escape, Or Is the Proof I Read Incorrect? This is a delightful problem I came across and took a long time to find a solution. Apparently it was incorrectly unsolved for 25 years.
"A man is stuck in a perfectly circular arena with a lion. The man can move as fast as the lion. Is it possible for the man to survive? (Assume each has infinite strength so they can both continue to move indefinitely if needed)."
I posted a detailed solution from a proof I read in a book. Apparently a lot of people disagree with this proof. I think it's a wonderful problem, was curious what you guys thought.
Writeup of solution
References: The post I wrote up followed a proof presented in this book: Famous Puzzles of Great Mathematicians. The author of that book said he based his proof on 2 papers.
How the Lion Tamer was Saved, by Richard Rado, Mathematical Spectrum Volume 6 (1973/1974). 
More About Lions and Other Animals, by Peter Rado and Richard Rado, Volume 7 (1974/1975). 
 A: According to the different links you and Presh provided, here is how I understand the solution :
Let d be the distance between the man and the lion, and let t be the time passing.
$$\lim\limits_{t \to \infty} d = 0$$
What does it mean ?
It means that the man can theoretically escape if we assimilate the man and the lion to points. However, if we consider that they are circles with non null diameter, the lion catches the man.
A: Thanks to a comment from "Blue," I came across this, and I feel that settles the matter as it confirms the references provided adequate proofs.
Wolfram Mathworld's entry on this problem states the man can survive. http://mathworld.wolfram.com/LionandManProblem.html

A lion and a man in a closed arena have equal maximum speeds. What
  tactics should the lion employ to be sure of his meal? This problem
  was stated by Rado in 1925 (Littlewood 1986).
An incorrect "solution" is for the lion to get onto the line joining
  the man to the center of the arena and then remaining at this radius
  however the man moves. Besicovitch showed the man had a path of
  safety, although the lion would come arbitrarily close.

