MathDB
MMO 309 Moscow MO 1955 numbering sides of n-gon

Source:

August 17, 2019
combinatorial geometrycombinatoricsnumber theorynumerationconvex polygon

Problem Statement

A point OO inside a convex nn-gon A1A2...AnA_1A_2 . . .A_n is connected with segments to its vertices. The sides of this nn-gon are numbered 11 to nn (distinct sides have distinct numbers). The segments OA1,OA2,...,OAnOA_1,OA_2, . . . ,OA_n are similarly numbered.
a) For n=9n = 9 find a numeration such that the sum of the sides’ numbers is the same for all triangles A1OA2,A2OA3,...,AnOA1A_1OA_2, A_2OA_3, . . . , A_nOA_1.
b) Prove that for n=10n = 10 there is no such numeration.
Back to ProblemsView on AoPS