I mean, there's two ways you can go. There's high school level problem solving texts and then there's college level math.

If they're looking for interesting high school level math problems, then I'd look into Art of Problem Solving series (the two texts at the bottom). There's also Problem Solving Strategies by Engel, which is also an excellent book. If they're looking into some accessible "higher level" (meaning, proof-based) math, I'd suggest number theory. Niven has an excellent treatment of introductory number theory.

As far as introductory college level math goes, you could always hand them a multivariable calculus, linear algebra or differential equations book. As far as recommendations go, Axler has a pretty good treatment of linear algebra if you already have some background, Apostol's series on calculus (vol. 1 and vol. 2) has a more theoretical approach to single and multi variable calculus (it's more advanced than high school calculus, but not quite an undergraduate analysis text), and Birkhoff/Rota's differential equations book has an advanced approach to differential equations probably not for high schoolers. If you don't think they'll be scared by rigor, you could always refer them to Little Rudin, which is pretty much the standard for an undergraduate real analysis class. If you're looking for an analysis book that isn't ridiculously overpriced, I've heard about this one but I've never actually read it. For a pretty readable treatment of algebra (a.k.a. modern algebra or abstract algebra), see Artin. For a more theoretical approach (less recommended for high schoolers I guess), see MacLane/Birkhoff.

That's because Runge-Kutta is a well known method. You can even find an algorithm on Wikipedia. For the speaker's audience, this is common knowledge - it's covered in basic differential equations classes (or numerical methods classes) - and it's boring to go over the details. People aren't going to his talk to revisit elementary concepts. If you don't know the basic concepts required to understand his talk and aren't even willing to look it up on Wikipedia, stop going to talks.