Difference between revisions of "2024 AIME II Problems/Problem 8"
Prof joker (talk | contribs) |
Prof joker (talk | contribs) |
||
Line 8: | Line 8: | ||
<asy> | <asy> | ||
− | unitsize(0. | + | unitsize(0.35cm); |
pair O = (0, 0); | pair O = (0, 0); | ||
real r1 = 11; | real r1 = 11; | ||
Line 46: | Line 46: | ||
<asy> | <asy> | ||
− | unitsize(0. | + | unitsize(0.35cm); |
pair O = (0, 0); | pair O = (0, 0); | ||
real r1 = 11; | real r1 = 11; | ||
Line 74: | Line 74: | ||
label("$E$", E, NE); | label("$E$", E, NE); | ||
label("$F_o$", F, SW); | label("$F_o$", F, SW); | ||
− | label("$G_o$", G, | + | label("$G_o$", G, S); |
label("$H_o$", H, W); | label("$H_o$", H, W); | ||
label("$r_o$", 0.5 * H + 0.5 * G, NE); | label("$r_o$", 0.5 * H + 0.5 * G, NE); |
Revision as of 10:53, 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 |
[[Category:]] The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.