  In this paper we prove that every subset of $\mathbb{F}_p^2$ meeting all
$p+1$ lines passing through the origin has a zero-sum subset. This is motivated
by a result of Gao, Ruzsa and Thangadurai which states that
$OL(\mathbb{F}_{p}^{2})=p+OL(\mathbb{F}_{p})-1$, for sufficiently large primes
$p$. Here $OL(G)$ denotes the so-called Olson constant of the additive group
$G$ and represents the smallest integer such that no subset of cardinality
$OL(G)$ is zero-sum-free. Our proof is in the spirit of the Combinatorial
Nullstellensatz.
