Some great mini lecture videos by Richard Rusczyk (from ArtofProblemSolving) for mathcounts.  You can view the video, and there are associated worksheets to try similar problems as well.  This is a link to the September 2011 video for algebra word problems.  You might want to try the problems first, and then see how he solved it. 

Number Theory Problems Collection
Math will Rock your World : Math is beneath almost every modern technological tool and business activity!
Math is Fun : Learning Math in a fun way with easily understood illustrations
Math Olympiad Problems for Practice

Canadian Open Mathematics Challenge :  Practice Questions and Solutions included
Math Drills for MATHCOUNTS :  Online Test Problems and solutions.

Contest Problems with Solutions




Search for Answers to Difficult Topics or Problems

Ask Dr Math (The Math Forum @ Drexel :  Many difficult math problems are explained. Be not surprised that you can find almost any Olympiad type problems using its search feature.  Just type in keywords of your problem eg. "maximum number of regions"; "odd number followed by two even numbers"

Wolfram Mathworld :  Find almost anything about math here (especially the harder to understand ones)