A: 

www.topcoder.com is one I've used.

joelt
+10  A: 

Project Euler is the place to go--lots of mathematical problems that have interesting computational solutions.

JSBangs
Ouch, scooped by 15 seconds while I wrote a little more text. Oh well.
Jefromi
thank you, it is really good site.
hey
A: 

This question should provide you with a nigh-infinite supply of fun problems.

I also recommend hiring puzzles, such as those from Facebook and ITA.

Kevin L.
A: 
  • Google CodeJam - There you can find tasks from previous contests

  • Spoj.Pl - have a lot of ongoing contests and tons of problems

Andrei Taptunov
A: 

Any of ACM online contests.

http://uva.onlinejudge.org/ for example.

Dmitry
+2  A: 

http://www.claymath.org/millennium/P_vs_NP/ (they even pay you for solving it). More of them here: http://www.claymath.org/millennium/

jkramer
A: 

I have the USSR Olympiad Problem Book. It is supposed to be solved by brain power (not computer power), but I have found some problems extremely fun to program after I've solved them using pencil and paper.

Many have worked the "Hanoi Tower" problem both in paper and computer, the book I recommend has more similar examples!

Beware: I'm not saying this is a computer related book, I'm just stating that some problems explained therein are fun to program as well.

Cheers,

J.

Arrieta