Difference between revisions of "2004 AIME I Problems/Problem 3"
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== Solution == | == Solution == | ||
− | 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, <math>60</math> are edges. Each [[triangle|triangular]] face has <math>0</math> face diagonals and each [[quadrilateral]] face has <math>2</math>, 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. | + | 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, <math>60</math> are edges. Each [[triangle|triangular]] face has <math>0</math> face diagonals and each [[quadrilateral]] face has <math>2</math>, so there are <math>2 \cdot 12 = 24</math> face diagonals. This leaves <math>325 - 60 - 24 = \boxed{241}</math> segments to be the space diagonals. |
== See also == | == See also == |
Latest revision as of 16:56, 1 January 2016
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 |
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