MathDB
Defining the sequence from neighbours of P

Source: Romanian TST 2001

January 16, 2011
functioncombinatorics unsolvedcombinatorics

Problem Statement

Consider a convex polyhedron PP with vertices V1,…,VpV_1,\ldots ,V_p. The distinct vertices ViV_i and VjV_j are called neighbours if they belong to the same face of the polyhedron. To each vertex VkV_k we assign a number vk(0)v_k(0), and construct inductively the sequence vk(n) (n≥0)v_k(n)\ (n\ge 0) as follows: vk(n+1)v_k(n+1) is the average of the vj(n)v_j(n) for all neighbours VjV_j of VkV_k . If all numbers vk(n)v_k(n) are integers, prove that there exists the positive integer NN such that all vk(n)v_k(n) are equal for n≥Nn\ge N .
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