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From a sequence, define other sequences

Source: Nordic MO 2004 Q3

April 20, 2013
algebra unsolvedalgebra

Problem Statement

Given a finite sequence x1,1,x2,1,…,xn,1x_{1,1}, x_{2,1}, \dots , x_{n,1} of integers (n≥2)(n\ge 2), not all equal, define the sequences x1,k,…,xn,kx_{1,k}, \dots , x_{n,k} by x_{i,k+1}=\frac{1}{2}(x_{i,k}+x_{i+1,k}) \text{where }x_{n+1,k}=x_{1,k}. Show that if nn is odd, then not all xj,kx_{j,k} are integers. Is this also true for even nn?
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