Let P be a regular polygon with 4k+1 sides (where k is a natural) whose vertices are A1,A2,...,A4k+1 (in that order ). Each vertex Aj of P is assigned a natural of the set {1,2,...,4k+1} such that no two vertices are assigned the same number. On P the following operation is performed: Let Bj be the midpoint of the side AjAj+1 for j=1,2,...,4k+1 (where is consider A4k+2=A1). If a, b are the numbers assigned to Aj and Aj+1, respectively, the midpoint Bj is written the number 7a−3b. By doing this with each of the 4k+1 sides, the 4k+1 vertices initially arranged are erased.We will say that a natural m is fatal if for all natural k , no matter how the vertices of P are initially arranged, it is impossible to obtain 4k+1 equal numbers through a finite amount of operations from m.a) Determine if the 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 m es fatal si no importa cómo se disponen inicialmente los vértices de P, es imposible obtener mediante una cantidad finita de operaciones 4k+1 números iguales a m. combinatoricsregular polygongeometrygeometric transformation