Rado Harman once asked a quite interesting question (sorry, only in Slovak) on his blog. Being as stubborn as I am, I didn't give up until I was pretty sure I found the full solution (the proof has been updated on 2012/02/05 to fix some grammar and style mistakes).
Another time, I stumbled upon the Unsolved Problems and Rewards webpage of Prof. Clark Kimberling. Most of the problems looked both nice and challenging. In the end, a few pages of my notebook became victims of problem 7. The Mysterious B Sequence… until the full picture emerged and the proof appeared in plain sight.
Some problems are more suited for computers than humans. Another Peter asked us to prove that 1847^2013+2 is indeed a prime. Since this number is of general form, I used the excellent Primo program by Marcel Martin and let it run in the spare time for a few days. The result? Yes, the number is now a certified prime (you can check the certificate yourself and it is considerably faster than searching for the certificate).
Apart from problem solving, I occasionally write a piece of text here or there on various subjects. One of them was a brief introduction to Banach spaces (Postscript format only) and their uses, aimed at students of Mathematical Analysis 3 (multivariable real analysis) at Faculty of Mathematics, Physics and Informatics of Comenius University.