A question which, I suppose, is as easy as abc for an expert. For a given finite field $F$ of odd characteristic (if needed, the characteristic is $3$ and the size of $F$ is large), do there exist two quadratic forms on $F^5$ without non-trivial common zeroes? More generally, what should be the relation between $n$ and $k$ in order for $k$ quadratic forms on $F^n$ without non-trivial common zeroes to exist?
Chevalley-Warning: If you have a system of forms of degrees $d_1,...,d_k$ in $n$ variables, they will have a common non-trivial zero if $n > \sum d_i$. For $d_i=2$, the condition is $n > 2k$.
There is a full proof in the wikipedia page: http://en.wikipedia.org/wiki/Chevalley%E2%80%93Warning_theorem