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numbering vertices of n-gon from 0 to n-1 (Chile NMO 1990 P3)

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November 20, 2021
combinatoricsArithmetic Progression

Problem Statement

Given a polygon with nn sides, we assign the numbers 0,1,...,n−10,1,...,n-1 to the vertices, and to each side is assigned the sum of the numbers assigned to its ends. The figure shows an example for n=5n = 5. Notice that the numbers assigned to the sides are still in arithmetic progression. https://cdn.artofproblemsolving.com/attachments/c/0/975969e29a7953dcb3e440884461169557f9a7.png ∙\bullet Make the respective assignment for a 99-sided polygon, and generalize for odd nn. ∙\bullet Prove that this is not possible if nn is even.
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