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2023 AMC 10A Problem 6

Problem 6 of 25EasierGeometryCombinatorics

An integer is assigned to each vertex of a cube. The value of an edge is defined to be the sum of the values of the two vertices it touches, and the value of a face is defined to be the sum of the values of the four edges surrounding it. The value of the cube is defined as the sum of the values of its six faces. Suppose the sum of the integers assigned to the vertices is 21.21. What is the value of the cube?

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Solution

Count by incidences. Each edge lies on 22 faces, so the six face values together are 22 times the total of all edge values. Each vertex lies on 33 edges, so the total edge value is 33 times the vertex sum. Chaining these, the cube’s value is 2⋅3⋅21=126.2 \cdot 3 \cdot 21 = 126. Therefore, the answer is D.
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Tagged: double counting · cube geometry

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