Difference between revisions of "2014 AIME I Problems/Problem 10"
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== Problem 10 == | == Problem 10 == | ||
| + | A disk with radius <math>1</math> is externally tangent to a disk with radius <math>5</math>. Let <math>A</math> be the point where the disks are tangent, <math>C</math> be the center of the smaller disk, and <math>E</math> be the center of the larger disk. While the larger disk remains fixed, the smaller disk is allowed to roll along the outside of the larger disk until the smaller disk has turned through an angle of <math>360^\circ</math>. That is, if the center of the smaller disk has moved to the point <math>D</math>, and the point on the smaller disk that began at <math>A</math> has now moved to point <math>B</math>, then <math>\overline{AC}</math> is parallel to <math>\overline{BD}</math>. Then <math>\sin^2(\angle BEA)=\tfrac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | ||
== Solution == | == Solution == | ||
Revision as of 19:55, 14 March 2014
Problem 10
A disk with radius 
is externally tangent to a disk with radius 
. Let 
be the point where the disks are tangent, 
be the center of the smaller disk, and 
be the center of the larger disk. While the larger disk remains fixed, the smaller disk is allowed to roll along the outside of the larger disk until the smaller disk has turned through an angle of 
. That is, if the center of the smaller disk has moved to the point 
, and the point on the smaller disk that began at has now moved to point 
, then 
is parallel to 
. Then 
, where 
and 
are relatively prime positive integers. Find 
.
Solution
See also
| 2014 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 9 |
Followed by Problem 11 | |
| 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. 
