Let n≥3 be an integer and let S⊂{1,2,…,n3} be a set with 3n2 elements. Prove that there exist nine distinct numbers a1,a2,…,a9∈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 functionalgebra proposedalgebrapigeonhole principle