RO-TST 1996 Problem 16

Let $n\geq 3$ be an integer and let $\mathcal{S} \subset \{1,2,\ldots, n^3\}$ be a set with $3n^2$ elements. Prove that there exist nine distinct numbers $a_1,a_2,\ldots,a_9 \in \mathcal{S}$ such that the following system has a solution in nonzero integers: \begin{eqnarray*} a_1x + a_2y +a_3 z &=& 0 a_4x + a_5 y + a_6 z &=& 0 a_7x + a_8y + a_9z &=& 0. \end{eqnarray*}

Marius Cavachi