For years I have visited the UVa Online Judge site to practice programming skills and just recently ( thanks to this site ) discovered the Sphere Online Judge site. I am interested to know which challenges from these sites folks here found the most interesting, difficult, satisfying, perplexing, or amusing.
I wouldn't say there is a problem that I like THE most, but I always love it when the problem is wrapped in a nice story, e.g. such as LA 4296 - Bring Your Own Horse
Personally, I like "Check the Check" (Problem 10196) -- mainly because I love chess and, as Steven Skiena has pointed out, once you solve this one you're well on your way to building a complete chess engine.
You may want to check out shygypsy.com/acm. He grades the problems by difficulty, so you can see where you stand. It's also interesting to see how many problems have essentially no solution -- (or at least, have not yet been solved by anyone on that site).
Personally, I need a more gentle introduction to algorithmic problems -- none of my degrees are in CS, and I'm playing self-education catch-up. For that reason, I prefer USACO's Training Site.
Edit: I forgot to finish the last sentence of the second paragraph. Fixed that.
I really like the TopCoder algorithm competitions and practice rooms.