I am tutoring several talented students, middle school level and early high school level, in mathematics. I am always looking for new sources from which to draw questions. Can anyone recommend books, web-sites, etc. with a interesting questions?
I know of the ArtofProblemSolving.com. I am currently using the books "Challenging Problems in Algebra" and "Challenging Problems in Geometry" by Posamentier and Salkind.
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There is this "Math Help Forum" which discusses about high school level and undergraduate level mathematics. You can find lots and lots of things. Here is one problem website : http://amc.maa.org/e-exams/e8-usamo/usamo.shtml
Calculus by Thomas finney is a Classic book for teaching Calculus.
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Mathcounts is aimed at talented middle school students. If you are affiliated with a school, you can register as a coach with them, and they can provide you with many helpful resources. If not, they still have problem sets available for sale from their catalog, as well as coaching handbooks, which include solutions and teaching advice.
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Check out the Blossoms videos. Although aimed more at a classroom setting there may be interesting snippets to grab.
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I have developed some educational freeware to teach students how to write simple, though thoroughly rigorous mathematical proofs. The software verifies every line of proof as it is entered. It also provides some important insights into the nature of modern mathematics. One of the worked examples in the tutorial is to prove the set of all things cannot exist. (It would be too big, and would include at least one very troublesome object that would make the whole thing blow up!) I think your budding mathematicians would enjoy it. Download the software and tutorial at my website: http://www.dcproof.com
If you haven't been using "The Art and Craft of Problem Solving" by Zeitz, you should. Like, right at this moment. :)
For neat problems... you might want to look into "Which Way Did The Bicycle Go?" by Konhauser, Wagon and Velleman.