2009 AMC 12B Problem 20

A convex polyhedron

Q

has vertices

V_1,

V_2,

\ldots,

V_n,

and

100

edges. The polyhedron is cut by planes

P_1,

P_2,

\ldots,

P_n

in such a way that plane

P_k

cuts only those edges that meet at vertex

V_k.

In addition, no two planes intersect inside or on

Q.

The cuts produce

n

pyramids and a new polyhedron

R.

How many edges does

R

have?

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Solution

Each of the

100

edges is cut once near each endpoint, so

R

has

2 \cdot 100 = 200

vertices. The cut at vertex

V_k

creates a small polygon whose number of edges equals the degree of

V_k

; summed over all vertices this is

200,

the total number of edge-endpoints. The middle portion of each original edge also survives, adding

100

edges. So

R

has

200 + 100 = 300

edges. Thus, the correct answer is C.