Difference between revisions of "2022 AIME I Problems/Problem 10"
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== Problem == | == Problem == | ||
Three spheres with radii <math>11</math>, <math>13</math>, and <math>19</math> are mutually externally tangent. A plane intersects the spheres in three congruent circles centered at <math>A</math>, <math>B</math>, and <math>C</math>, respectively, and the centers of the spheres all lie on the same side of this plane. Suppose that <math>AB^2 = 560</math>. Find <math>AC^2</math>. | Three spheres with radii <math>11</math>, <math>13</math>, and <math>19</math> are mutually externally tangent. A plane intersects the spheres in three congruent circles centered at <math>A</math>, <math>B</math>, and <math>C</math>, respectively, and the centers of the spheres all lie on the same side of this plane. Suppose that <math>AB^2 = 560</math>. Find <math>AC^2</math>. | ||
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| + | ==solution 1== | ||
| + | Let the distance between the center of the sphere to the center of those circular intersections as <math>a,b,c</math> separately. <math>a-11,b-13,c-19</math>. According to the problem, we have <math>a^2-11^2=b^2-13^2=c^2-19^2;(11+13)^2-(b-a)^2=560</math>. After solving we have <math>b-a=4</math>, plug this back to <math>11^2-a^2=13^2-b^2; a=4,b=8,c=16</math> | ||
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| + | The desired value is <math>(11+19)^2-(16-4)^2=\boxed{756}</math> | ||
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| + | ~bluesoul | ||
==See Also== | ==See Also== | ||
{{AIME box|year=2022|n=I|num-b=9|num-a=11}} | {{AIME box|year=2022|n=I|num-b=9|num-a=11}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 23:02, 17 February 2022
Problem
Three spheres with radii 
, 
, and 
are mutually externally tangent. A plane intersects the spheres in three congruent circles centered at 
, 
, and 
, respectively, and the centers of the spheres all lie on the same side of this plane. Suppose that 
. Find 
.
solution 1
Let the distance between the center of the sphere to the center of those circular intersections as 
separately. 
. According to the problem, we have 
. After solving we have 
, plug this back to 
The desired value is 
~bluesoul
See Also
| 2022 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. 
