Fermat's last theorem fails in $\mathbb{Z}/p\mathbb Z$ for $p$ sufficiently large 
Statement 
For any $n, \;x^n+y^n=z^n$ has non-trivial solutions in $\mathbb{Z}/p\mathbb Z$ for all but finitely many $p$.

I remember seeing this problem on an first year undergraduate problem sheet, but never succeeded in solving it. I cannot find an elementary solution on the internet though: does anyone know of one?

Not sure what you consider to be elementary but it can be solved with Schur's theorem. See:
http://math.mit.edu/~fox/MAT307-lecture05.pdf
It is theorem 4 in that paper.
