Interesting "real life" applications of serious theorems As student in mathematics, one sometimes encounters exercises which ask you to solve a rather funny "real life problem", e.g. I recall an exercise on the Krein-Milman theorem which was something like: 
"You have a great circular pizza with $n$ toppings. Show that you can divide the pizza equitably among $k$ persons, which means every person gets a piece of pizza with exactly $\frac{1}{k}$ of any of the $n$ topping on it." 
Are there more examples which are particular interesting or instructive?
EDIT: Since this is turning into a list of mathematical jokes or sophisticated proofs for simple facts, I may have to be more precise what I was asking for: a "real life example didactically used to motivate a mathematical theorem" (thanks to Lord_Gestalter for this great wording).

Euler's Theorem forms the basis of the RSA encryption system that's used in virtually every form of web-based cryptography (most notably, SSL and TLS protocols). "A Practical Application For Prime Numbers" provides an explanation better than I ever could. Regardless, the practical implications of this theorem are vast, and widely used (even right now, as I am about to submit this reply).
/I'm not a mathematician, but felt compelled to respond.
