2004 AIME I Problems/Problem 3
Problem
A convex polyhedron has vertices, edges, and faces, of which are triangular and 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, are edges. Each triangular face has face diagonals and each quadrilateral face has , so there are face diagonals. This leaves segments to be the space diagonals.
See also
2004 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
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