AT-OEMO-BEG 2018 Problem 3

For a given integer $n \ge 4$ we examine whether there exists a table with three rows and $n$ columns which can be filled by the numbers $1, 2,...,, 3n$ such that $\bullet$ each row totals to the same sum $z$ and $\bullet$ each column totals to the same sum $s$ . Prove: (a) If $n$ is even, such a table does not exist. (b) If $n = 5$ , such a table does exist.

(Gerhard J. Woeginger)