Interesting “real life” applications of serious theorems [closed]

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 1k 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).


n2 is not rational for n3

If n2=pq then qn+qn=pn contradicting Fermat’s last theorem.

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