Prove that 'pi' is irrational.
This is not a one-liner proof, but also not so bad. The trickier part is to prove that pi is transcendental. (i.e. not a root of any polynomial with integer coefficients)
There's a relatively simple proof of the irrationality of pi up at:
http://pi314.at/math/irrational.html
It is a contradiction proof due to Niven. Not exactly the sort of thing one would come up with off the top of one's head, but not so hard to understand.
The solution if people would like to see it.
