Difference between revisions of "1960 IMO Problems/Problem 6"
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==Problem== | ==Problem== | ||
| + | Consider a cone of revolution with an inscribed sphere tangent to the base of the cone. A cylinder is circumscribed about this sphere so that one of its bases lies in the base of the cone. let <math>V_1</math> be the volume of the cone and <math>V_2</math> be the volume of the cylinder. | ||
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| + | a) Prove that <math>V_1 \neq V_2</math>; | ||
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| + | b) Find the smallest number <math>k</math> for which <math>V_1 = kV_2</math>; for this case, construct the angle subtended by a diamter of the base of the cone at the vertex of the cone. | ||
==Solution== | ==Solution== | ||
Revision as of 10:51, 28 October 2007
Problem
Consider a cone of revolution with an inscribed sphere tangent to the base of the cone. A cylinder is circumscribed about this sphere so that one of its bases lies in the base of the cone. let 
be the volume of the cone and 
be the volume of the cylinder.
a) Prove that 
;
b) Find the smallest number 
for which 
; for this case, construct the angle subtended by a diamter of the base of the cone at the vertex of the cone.
Solution
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See Also
| 1960 IMO (Problems) • Resources | ||
| Preceded by Problem 5 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Final Question |
| All IMO Problems and Solutions | ||
