Difference between revisions of "2024 AIME II Problems/Problem 8"
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Torus <math>T</math> is the surface produced by revolving a circle with radius 3 around an axis in the plane of the circle that is a distance 6 from the center of the circle (so like a donut). Let <math>S</math> be a sphere with a radius 11. When <math>T</math> rests on the outside of <math>S</math>, it is externally tangent to <math>S</math> along a circle with radius <math>r_i</math>, and when <math>T</math> rests on the outside of <math>S</math>, it is externally tangent to <math>S</math> along a circle with radius <math>r_o</math>. The difference <math>r_i-r_o</math> can be written as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | Torus <math>T</math> is the surface produced by revolving a circle with radius 3 around an axis in the plane of the circle that is a distance 6 from the center of the circle (so like a donut). Let <math>S</math> be a sphere with a radius 11. When <math>T</math> rests on the outside of <math>S</math>, it is externally tangent to <math>S</math> along a circle with radius <math>r_i</math>, and when <math>T</math> rests on the outside of <math>S</math>, it is externally tangent to <math>S</math> along a circle with radius <math>r_o</math>. The difference <math>r_i-r_o</math> can be written as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | ||
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| + | ==See also== | ||
| + | {{AIME box|year=2024|num-b=7|num-a=9|n=II}} | ||
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| + | [[Category:]] | ||
| + | {{MAA Notice}} | ||
Revision as of 20:29, 8 February 2024
Torus 
is the surface produced by revolving a circle with radius 3 around an axis in the plane of the circle that is a distance 6 from the center of the circle (so like a donut). Let 
be a sphere with a radius 11. When rests on the outside of , it is externally tangent to along a circle with radius 
, and when rests on the outside of , it is externally tangent to along a circle with radius 
. The difference 
can be written as 
, where 
and 
are relatively prime positive integers. Find 
.
See also
| 2024 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 7 |
Followed by Problem 9 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
[[Category:]]
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
