2009 AMC 12B Problem 20
Attempt Problem 20 of the 2009 AMC 12B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2009 AMC 12B solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
20.
A convex polyhedron has vertices and 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: C
Small hint:
Each original edge is cut once near each of its two endpoints, creating new vertices
Big hint:
Count the new edges (one per edge-endpoint) plus the surviving middle portion of each original edge
Solution:
Each of the edges is cut once near each endpoint, so has vertices.
The cut at vertex creates a small polygon whose number of edges equals the degree of ; summed over all vertices this is the total number of edge-endpoints. The middle portion of each original edge also survives, adding edges. So has edges.
Thus, the correct answer is C.
Problem 20 in other years
- 1950 AMC 12
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