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Given a finite sequence $x_{1,1}, x_{2,1}, \ldots, x_{n, 1}$ of integers ( $n \geq 2$ ), not all equal, define the sequences $x_{1, k}, \ldots, x_{n, k}$ by
$x_{i, k+1}=\frac{1}{2}\left(x_{i, k}+x_{i+1, k}\right), \quad \text { where } x_{n+1, k}=x_{1, k} .$
Show that if $n$ is odd, then not all $x_{j, k}$ are integers. Is this also true for even $n$ ?
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