Difference between revisions of "2022 AIME I Problems/Problem 10"

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draw(circle(OB,13));
 
draw(circle(OB,13));
 
draw(circle(OA,11));
 
draw(circle(OA,11));
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draw((-24,0)--(24,0));
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label("$l$",(-22,1),N);
  
 
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Revision as of 13:19, 21 February 2022

Problem

Three spheres with radii $11$, $13$, and $19$ are mutually externally tangent. A plane intersects the spheres in three congruent circles centered at $A$, $B$, and $C$, respectively, and the centers of the spheres all lie on the same side of this plane. Suppose that $AB^2 = 560$. Find $AC^2$.

Diagrams

[asy]  size(250); pair A, B, OA, OB;  B = (0,0); A = (-23.6643191,0); OB = (0,-8); OA = (-23.6643191,-4);  draw(circle(OB,13)); draw(circle(OA,11));  draw((-24,0)--(24,0)); label("$l$",(-22,1),N);  [/asy]

Solution 1

Let the distance between the center of the sphere to the center of those circular intersections as $a,b,c$ separately. $a-11,b-13,c-19$. According to the problem, we have $a^2-11^2=b^2-13^2=c^2-19^2;(11+13)^2-(b-a)^2=560$. After solving we have $b-a=4$, plug this back to $11^2-a^2=13^2-b^2; a=4,b=8,c=16$

The desired value is $(11+19)^2-(16-4)^2=\boxed{756}$

~bluesoul

Solution 2

Denote by $r$ the radius of three congruent circles formed by the cutting plane. Denote by $O_A$, $O_B$, $O_C$ the centers of three spheres that intersect the plane to get circles centered at $A$, $B$, $C$, respectively.

Because three spheres are mutually tangent, $O_A O_B = 11 + 13 = 24$, $O_A O_C = 11 + 19 = 32$.

We have $O_A A^2 = 11^2 - r^2$, $O_B B^2 = 13^2 - r^2$, $O_C C^2 = 19^2 - r^2$.

Because $O_A A$ and $O_B B$ are perpendicular to the plane, $O_A AB O_B$ is a right trapezoid, with $\angle O_A A B = \angle O_B BA = 90^\circ$.

Hence, \begin{align*} O_B B - O_A A & = \sqrt{O_A O_B^2 - AB^2} \\ & = 4 . \hspace{1cm} (1) \end{align*}

Recall that \begin{align*} O_B B^2 - O_A A^2 & = \left( 13^2 - r^2 \right) - \left( 11^2 - r^2 \right) \\ & = 48 . \hspace{1cm} (2) \end{align*}

Hence, taking $\frac{(2)}{(1)}$, we get \[ O_B B + O_A A = 12 . \hspace{1cm} (3) \]

Solving (1) and (3), we get $O_B B = 8$ and $O_A A = 4$.

Thus, $r^2 = 11^2 - O_A A^2 = 105$.

Thus, $O_C C = \sqrt{19^2 - r^2} = 16$.

Because $O_A A$ and $O_C C$ are perpendicular to the plane, $O_A AC O_C$ is a right trapezoid, with $\angle O_A A C = \angle O_C CA = 90^\circ$.

Therefore, \begin{align*} AC^2 & = O_A O_C^2 - \left( O_C C - O_A A \right)^2 \\ & = \boxed{\textbf{(756) }} . \end{align*}

$\textbf{FINAL NOTE:}$ In our solution, we do not use the conditio that spheres $A$ and $B$ 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 (ProblemsAnswer KeyResources)
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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