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
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 2 (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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