Problem:
A convex polyhedron has vertices , and 100 edges. The polyhedron is cut by planes in such a way that plane cuts only those edges that meet at vertex . In addition, no two planes intersect inside or on . The cuts produce pyramids and a new polyhedron . How many edges does have?
Answer Choices:
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Solution:
Each edge of is cut by two planes, so has 200 vertices. Three edges of meet at each vertex, so has edges.
OR
At each vertex, as many new edges are created by this process as there are original edges meeting that vertex. Thus the total number of new edges is the total number of endpoints of the original edges, which is 200 . A middle portion of each original edge is also present in , so has edges.
OR
Euler's Polyhedron Formula applied to gives , where is the number of faces of . Each edge of is cut by two planes, so has 200 vertices. Each cut by a plane creates an additional face on , so Euler's Polyhedron Formula applied to gives , where is the number of edges of . Subtracting the first equation from the second gives , so .
The problems on this page are the property of the MAA's American Mathematics Competitions