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fatal numbers , numbers on vertices of regular 4k+1 polygon

Source: OIFMAT I 2010 day 1 p3 - Chilean Math Forum FMAT Olympiad https://artofproblemsolving.com/community/c2484778_oifmat

September 25, 2021
combinatoricsregular polygongeometrygeometric transformation

Problem Statement

Let PP be a regular polygon with 4k+1 4k + 1 sides (where k k is a natural) whose vertices are A1,A2,...,A4k+1 A_1, A_2, ..., A_ {4k + 1} (in that order ). Each vertex Aj A_j of PP is assigned a natural of the set {1,2,...,4k+1} \{1,2, ..., 4k + 1 \} such that no two vertices are assigned the same number. On PP the following operation is performed: Let Bj B_j be the midpoint of the side AjAj+1 A_jA_ {j + 1} for j=1,2,...,4k+1 j = 1,2, ..., 4k + 1 (where is consider A4k+2=A1 A_ {4k + 2} = A_1 ). If a a , b b are the numbers assigned to Aj A_ {j} and Aj+1 A_ {j + 1} , respectively, the midpoint Bj B_j is written the number 7a−3b 7a-3b . By doing this with each of the 4k+1 4k + 1 sides, the 4k+1 4k + 1 vertices initially arranged are erased.
We will say that a natural m m is fatal if for all natural k k , no matter how the vertices of PP are initially arranged, it is impossible to obtain 4k+1 4k + 1 equal numbers through a finite amount of operations from m m .
a) Determine if the 2010 2010 is fatal or not. Justify. b) Prove that there are infinite fatal numbers.
[color=#f00]PS. A help in translation of the 2nd paragraph is welcome. [hide=Original wording]Diremos que un natural mm es fatal si no importa cómo se disponen inicialmente los vértices de P{P}, es imposible obtener mediante una cantidad finita de operaciones 4k+14k+1 números iguales a mm.
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