Assigning numbers on the edges of a polyhedron
Source: 2024 Vietnam National Olympiad - Problem 7
January 6, 2024
number theorycombinatorics
Problem Statement
In the space, there is a convex polyhedron such that for every vertex of , there are an even number of edges passing through that vertex. We choose a face of . Then we assign each edge of a positive integer such that for all faces of different from , the sum of the numbers assigned on the edges of that face is a positive integer divisible by . Prove that the sum of the numbers assigned on the edges of is also a positive integer divisible by .