Difference between revisions of "2009 AMC 12B Problems/Problem 20"
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Revision as of 09:57, 4 July 2013
Problem
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?
Solution
Solution 1
Each edge of is cut by two planes, so has 
vertices. Three edges of meet at each vertex, so has 
edges.
Solution 2
At each vertex, as many new edges are created by this process as there are original edges meeting at that vertex. Thus the total number of new edges is the total number of endpoints of the original edges, which is . A middle portion of each original edge is also present in , so has 
edges.
Solution 3
Euler's Polyhedron Formula applied to gives 
, where F is the number of faces of . Each edge of is cut by two planes, so has 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 
, whence 
. The answer is 
.
See also
| 2009 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 19 |
Followed by Problem 21 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
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