Difference between revisions of "2024 AIME II Problems/Problem 8"
Prof joker (talk | contribs) |
Prof joker (talk | contribs) |
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draw(F--E, dashed); | draw(F--E, dashed); | ||
draw(G--H, dashed); | draw(G--H, dashed); | ||
− | label(" | + | label("$O$", O, SW); |
− | label(" | + | label("$A$", A, SW); |
− | label(" | + | label("$B$", B, NW); |
− | label(" | + | label("$C$", C, NW); |
− | label(" | + | label("$D$", D, SW); |
− | label(" | + | label("$E_i$", E, NE); |
− | label(" | + | label("$F_i$", F, W); |
− | label(" | + | label("$G_i$", G, SE); |
− | label(" | + | label("$H_i$", H, W); |
− | label(" | + | label("$r_i$", 0.5 * H + 0.5 * G, NE); |
− | label(" | + | label("$3$", 0.5 * E + 0.5 * G, NE); |
− | label(" | + | label("$11$", 0.5 * O + 0.5 * G, NE); |
− | < | + | </asy> |
<asy> | <asy> | ||
Line 65: | Line 65: | ||
draw(F--E, dashed); | draw(F--E, dashed); | ||
draw(G--H, dashed); | draw(G--H, dashed); | ||
− | label(" | + | label("$O$", O, SW); |
− | label(" | + | label("$A$", A, SW); |
− | label(" | + | label("$B$", B, NW); |
− | label(" | + | label("$C$", C, NW); |
− | label(" | + | label("$D$", D, SW); |
− | label(" | + | label("$E$", E, NE); |
− | label(" | + | label("$F_o$", F, SW); |
− | label(" | + | label("$G_o$", G, N); |
− | label(" | + | label("$H_o$", H, W); |
− | label(" | + | label("$r_o$", 0.5 * H + 0.5 * G, NE); |
− | label(" | + | label("$3$", 0.5 * E + 0.5 * G, NE); |
− | label(" | + | label("$11$", 0.5 * O + 0.5 * G, NE); |
− | < | + | </asy> |
For both graphs, point <math>O</math> is the center of sphere <math>S</math>, and points <math>A</math> and <math>B</math> are the intersections of the sphere and the axis. Point <math>E</math> (ignoring the subscripts) is one of the circle centers of the intersection of torus <math>T</math> with section <math>\mathcal{P} </math>. Point <math>G</math> (again, ignoring the subscripts) is one of the tangents between the torus <math>T</math> and sphere <math>S</math> on section <math>\mathcal{P} </math>. <math>EF\bot CD</math>, <math>HG\bot CD</math>. | For both graphs, point <math>O</math> is the center of sphere <math>S</math>, and points <math>A</math> and <math>B</math> are the intersections of the sphere and the axis. Point <math>E</math> (ignoring the subscripts) is one of the circle centers of the intersection of torus <math>T</math> with section <math>\mathcal{P} </math>. Point <math>G</math> (again, ignoring the subscripts) is one of the tangents between the torus <math>T</math> and sphere <math>S</math> on section <math>\mathcal{P} </math>. <math>EF\bot CD</math>, <math>HG\bot CD</math>. |
Revision as of 11:43, 9 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
.
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
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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The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.