2022 AIME I Problems/Problem 10
Contents
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 .
Diagram
~MRENTHUSIASM
Solution 1
This solution refers to the Diagram section.
We let be the plane that passes through the spheres and and be the centers of the spheres with radii and . We take a cross-section that contains and , which contains these two spheres but not the third, as shown below: Because the plane cuts out congruent circles, they have the same radius and from the given information, . Since is a trapezoid, we can drop an altitude from to to create a rectangle and triangle to use Pythagorean theorem. We know that the length of the altitude is and let the distance from to be . Then we have .
We have because of the rectangle, so . Squaring, we have . Subtracting, we get . We also notice that since we had means that and since we know that , .
We take a cross-section that contains and , which contains these two spheres but not the third, as shown below: We have . Since , we have . Using Pythagorean theorem, . Therefore, .
~KingRavi
Solution 2
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 we have and
The desired value is
~bluesoul
Solution 3
Denote by the radius of three congruent circles formed by the cutting plane. Denote by , , the centers of three spheres that intersect the plane to get circles centered at , , , respectively.
Because three spheres are mutually tangent, , .
We have , , .
Because and are perpendicular to the plane, is a right trapezoid, with .
Hence,
Recall that
Hence, taking , we get
Solving (1) and (3), we get and .
Thus, .
Thus, .
Because and are perpendicular to the plane, is a right trapezoid, with .
Therefore,
In our solution, we do not use the conditio that spheres and are externally tangent. This condition is redundant in solving this problem.
~Steven Chen (www.professorcheneeu.com)
Video Solution
https://www.youtube.com/watch?v=SqLiV2pbCpY&t=15s
~Steven Chen (www.professorcheneeu.com)
Video Solution (Mathematical Dexterity)
https://www.youtube.com/watch?v=HbBU13YiopU
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 |
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