MathLabs

Problem 1

We call a 55-tuple of integers arrangeable if its elements can be labeled a,b,c,d,ea,b,c,d,e in some order so that a−b+c−d+e=29a-b+c-d+e=29. Determine all 20172017-tuples of integers n1,n2,…,n2017n_1,n_2,\ldots,n_{2017} such that if we place them in a circle in clockwise order, then any 55-tuple of numbers in consecutive positions on the circle is arrangeable.
Step 1 of 4: Shift by 2929
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mi:=ni−29,a−b+c−d+e=0m_i:=n_i-29,\quad a-b+c-d+e=0
Detailed analysis

Set mi=ni−29m_i=n_i-29 for each ii (indices mod 20172017). Since a−b+c−d+e=29a-b+c-d+e=29 can be rewritten as (a−29)−(b−29)+(c−29)−(d−29)+(e−29)=0(a-29)-(b-29)+(c-29)-(d-29)+(e-29)=0, every 55 consecutive mim_i's can be relabeled a,b,c,d,ea,b,c,d,e with a−b+c−d+e=0a-b+c-d+e=0. It suffices to show every mi=0m_i=0.