Where should I start with advanced mathematics during my holiday break?

[recon]

I'm in the middle of my three-month holiday and would like to have an assignment in mathematics, at the post-secondary/pre-undergraduate level. I already know basic trigonometry identities, polynomials (factor and remainder theorems) and very basic calculus (I am stumped by partial differential equations). I'm above average in my mathematics class, and have access to decaying undergraduate mathematics books (full of bookworms, yech) that were handed down to me by a friend. Those books cover almost everything: number theory, Fourier series, differential equations, matrix theory, numerical analysis, etc. I have not started on these books, because I simply don't know WHERE to start.

I'm willing to sacrifice the next one month of my holidays for this assignment, so make it TOUGH, but still accomplishable within at least 2 months. I don't have a preference for an area in mathematics. I consider myself to be a fairly fast learner, and I like it when I know what I'm working towards.

Oh, I'm 15 years old, but am more 16/17 than 15, being the youngest in my class.

 

[Hurkyl]

Here's a more exploratory problem:

What facts about polynomials are still true when you're using a different arithmetic system? One example is for all of your arithmetic to be done modulo 3.

Here are some interesting examples to ponder:

When you're doing arithmetic modulo 3, what do you think about the polynomial x^3 - x?

When you're doing arithmetic modulo 8, what are the factors of the polynomial x^2 - 1?

 

[FulhamFan3]

Have you solved a second order differential equation yet? If you haven't look into all steps needed to solve those.

Also look at LaPlace transforms.