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?