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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).

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This doesn't appear to be a question. – Thomas Andrews Jun 28 '13 at 15:21
"Apparently a lot of people disagree with this proof. I [...] was curious what you guys thought.". I think, this is a question, in other words: "Do you agree with this proof?" – Tomas Jun 28 '13 at 15:27
Eliding "I think it is a wonderful problem, " before "was curious what you guys thought" obviously clarifies what he wrote. But he didn't write that. @Tomas – Thomas Andrews Jun 28 '13 at 15:49
Okay, I guess I just got it the other way. – Tomas Jun 28 '13 at 15:56
It’s perfectly clear what is being asked. What’s the point of this petty nitpicking? – Brian M. Scott Jun 28 '13 at 20:48

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.

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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.

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.

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What does it mean to "remain at this radius"? On its face, it would mean staying at the same distance from the centre, which is pointless... – DJohnM Jun 29 '13 at 22:04

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