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).

The $n=2$ case of the Borsuk-Ulam theorem can be visualized by saying there exists some pair of antipodal points on the Earth with equal temperatures and barometric pressures.  Of course, this is assuming that temperature and pressure vary continuously.
Ramsey's theorem says that, if given a sufficiently large complete graph that has been arbitrarily colored with $n$ colors, then one can find a monochromatic complete subgraph of a particular size.  One example is as follows:  Given any $2$-coloring on $K_6$, then we are guaranteed to find a monochromatic subgraph of size $3$.  This has an interesting real-life interpretation: If we invite $6$ people to a party, then at least $3$ of them must be mutual acquaintances, or at least $3$ of them must be mutual strangers.
