Sum of reciprocal squares

There are two parts to the problem

1) Show that $\frac{\pi}{4} \gt \sqrt{2-\sqrt{2}}$

2) Show that $\sum_{n=2}^{\infty} \frac{1}{n^2} \gt \frac{3}{5}$

Incidentally 2) implies 1) and is a stronger result than 1) which has an easier proof.

[Solution]

Lion and Lion Tamer

There is a circular cage which has the lion and a lion tamer (assume point masses) in there somewhere. Both the lion and the tamer run at the same speed.

Can the lion catch the lion tamer? (In finite time).


Solution (Highlight to view):

This is a pretty hard problem and the surprising answer is no!

It seems like the lion should be able to catch the lion tamer by directly moving towards lion tamer, but the lion tamer can escape! This is assuming they are moving points in the 2D plane. The lion can get arbitrarily close, but cannot coincide.


This problem was proposed by Richard Rado in 1920s and was solved by Abram Besikovitch in 1950s.

A solutions appears in the the book "A Mathematical Miscellany" by Littlewood.

I don't have good links, so this might be a starting point: lion and man problem.