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).

This is a fairly simple, but interesting application. 
The mean-value theorem is used in some Automatic number plate recognition systems, based on average speed measurement. Using Optical character recognition, a computer system reads car plates at several fixed points along a road, and then uses the distance between these points to compute the car's average speed. By the mean-value theorem, the car must have gone at precisely that speed somewhere between the fixed points. A penalty can be issued accordingly, if this average speed exceeds the speed limit. 
The system is used in several countries. It is not allowed in California, via Section 40801, Speed Trap Prohibition, of the California Vehicle Code, that reads

No peace officer or other person shall use a speed trap in arresting, or participating or assisting in the arrest of, any person for any alleged violation of this code nor shall any speed trap be used in securing evidence as to the speed of any vehicle for the purpose of an arrest or prosecution under this code. 

