Did you know that according to MathWorld you can square the circle using the compass and straight-edge in the hyperbolic plane? How long have you known this? Why have you been keeping it from me?
MathForge links to an article on Hilbert’s “24th” problem. Hilbert, in his famous 1900 speech, proposed twenty-three open problems in mathematics, but apparently there was a twenty-fourth that he dropped from the list, to formalize the notion of simplicity of proofs, and prove that theorems have a unique simplest proof. Like many of Hilbert’s twenty-three, this is less a problem and more an open-ending research program.
Herbert Wilf has put several of his books online for download. I particularly recommend generatingfunctionology, which is an excellent account of the uses of generating functions in combinatorics. I regarded generating functions as a sleazy trick before I read that book. A=B, a book he co-wrote with Petrovsek and Zeilberger on combinatorial sums, is also very interesting. It turns out there is an eminently-implementable algorithm that will show, for a large class of formulas, when two such sums are equal.