Difference between revisions of "2004 AIME I Problems/Problem 3"
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Revision as of 14:49, 18 August 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.
