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.
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).