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 <math>3</math> around an axis in the plane of the circle that is a distance <math>6</math> from the center of the circle (so like a donut). Let <math>S</math> be a sphere with a radius <math>11</math>. When <math>T</math> rests on the | + | Torus <math>T</math> is the surface produced by revolving a circle with radius <math>3</math> around an axis in the plane of the circle that is a distance <math>6</math> from the center of the circle (so like a donut). Let <math>S</math> be a sphere with a radius <math>11</math>. When <math>T</math> rests on the inside of <math>S</math>, it is internally 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>\tfrac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. |
<asy> | <asy> |
Revision as of 16:29, 21 February 2024
Contents
Problem
Torus is the surface produced by revolving a circle with radius
around an axis in the plane of the circle that is a distance
from the center of the circle (so like a donut). Let
be a sphere with a radius
. When
rests on the inside of
, it is internally 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
.
Solution 1
First, let's consider a section of the solids, along the axis.
By some 3D-Geomerty thinking, we can simply know that the axis crosses the sphere center. So, that is saying, the
we took crosses one of the equator of the sphere.
Here I drew two graphs, the first one is the case when is internally tangent to
,
and the second one is when is externally tangent to
.
For both graphs, point is the center of sphere
, and points
and
are the intersections of the sphere and the axis. Point
(ignoring the subscripts) is one of the circle centers of the intersection of torus
with section
. Point
(again, ignoring the subscripts) is one of the tangents between the torus
and sphere
on section
.
,
.
And then, we can start our calculation.
In both cases, we know .
Hence, in the case of internal tangent, .
In the case of external tangent, .
Thereby, . And there goes the answer,
~Prof_Joker
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
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 |
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