Difference between revisions of "2004 AIME I Problems/Problem 3"
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Revision as of 19:00, 4 July 2013
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 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
