Difference between revisions of "2004 AIME I Problems/Problem 3"
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== Problem == | == Problem == | ||
| − | A [[convex]] | + | A [[convex]] polyhedron <math> P </math> has 26 vertices, 60 edges, and 36 faces, 24 of which are triangular and 12 of which are quadrilaterals. A space diagonal is a line segment connecting two non-adjacent vertices that do not belong to the same face. How many space diagonals does <math> P </math> have? |
== Solution == | == Solution == | ||
| − | Every pair of vertices of the polyhedron determines either an edge, a face diagonal or a space diagonal. We have <math>{26 \choose 2} = \frac{26\cdot25}2 = 325</math> total | + | Every pair of [[vertex | vertices]] of the [[polyhedron]] determines either an [[edge]], a face [[diagonal]] or a space diagonal. We have <math>{26 \choose 2} = \frac{26\cdot25}2 = 325</math> total [[line segment]]s determined by the vertices. Of these, 60 are edges. Each [[triangle|triangular]] face has 0 face diagonals and each [[quadrilateral]] face has 2, so there are <math>2 \cdot 12 = 24</math> face diagonals. This leaves <math>325 - 60 - 24 = 241</math> segments to be the space diagonals. |
== See also == | == See also == | ||
* [[2004 AIME I Problems]] | * [[2004 AIME I Problems]] | ||
Revision as of 11:42, 29 July 2006
Problem
A convex polyhedron 
has 26 vertices, 60 edges, and 36 faces, 24 of which are triangular and 12 of which are quadrilaterals. A space diagonal is a line segment connecting two non-adjacent vertices that do not belong to the same face. How many space diagonals does have?
Solution
Every pair of vertices of the polyhedron determines either an edge, a face diagonal or a space diagonal. We have 
total line segments determined by the vertices. Of these, 60 are edges. Each triangular face has 0 face diagonals and each quadrilateral face has 2, so there are 
face diagonals. This leaves 
segments to be the space diagonals.
