Determine all sequences
![(x_1,x_2,\ldots,x_{2011})](/media/m/8/d/3/8d3fb17dc78ff9662cf2fbff9ffc8c60.png)
of positive integers, such that for every positive integer
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
there exists an integer
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
with
![\sum^{2011}_{j=1} j x^n_j = a^{n+1} + 1](/media/m/5/b/0/5b0cf49e5cd26cdd0d30f5cc6133ca24.png)
Proposed by Warut Suksompong, Thailand
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Determine all sequences $(x_1,x_2,\ldots,x_{2011})$ of positive integers, such that for every positive integer $n$ there exists an integer $a$ with $$\sum^{2011}_{j=1} j x^n_j = a^{n+1} + 1$$
Proposed by Warut Suksompong, Thailand