NMC 2004 Problem 3

Let $x_{11}, x_{21}, \ldots, x_{n 1}, n>2$, be a sequence of integers. We assume that all of the numbers $x_{i 1}$ are not equal. Assuming that the numbers $x_{1 k}, x_{2 k}, \ldots, x_{n k}$ have been defined, we set

$$\begin{aligned} x_{i, k+1} & =\frac{1}{2}\left(x_{i k}+x_{i+1, k}\right), i=1,2, \ldots, n-1 \\ x_{n, k+1} & =\frac{1}{2}\left(x_{n k}+x_{1 k}\right) \end{aligned}$$

Show that for $n$ odd, $x_{j k}$ is not an integer for some $j, k$. Does the same conclusion hold for $n$ even?