Question

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?

$200$

$2n$

$300$

$400$

$4n$

Answer: C

Concepts:polyhedron double counting

Difficulty rating: 2040

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.

2009 AMC 12B Problem 20

Answer: C

Difficulty rating: 2040

Solution:

Problem 20 in Other Years