Fix integers $m \geq 3$ and $n \geq 3$. Each cell of an array with $m$ rows and $n$ columns is coloured one of two colours such that: (1) Both colours occur on every column; and (2) On every two rows the cells on the same column share colour on exactly $k$ columns. Show that, if $m$ is odd, then \[\frac{n(m-1)}{2m}\leq k\leq \frac{n(m-2)}{m}\] The Problem Selection Committee