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

m (→‎Solution 3)
(Rearranged the diagrams, and made the reference diagram 3D.)
Line 2: Line 2:
 
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>.
  
−
== Diagrams ==
+
== Diagram ==
 +
<asy>
 +
/* Made by MRENTHUSIASM */
 +
size(250);
 +
import graph3;
 +
import solids;
 +
 
 +
currentprojection=orthographic((7,0.2,2));
 +
triple A, B, C, OA, OB, OC;
 +
A = (0,0,0);
 +
B = (0,sqrt(560),0);
 +
C = intersectionpoints(Circle(A,sqrt(756),(0,0,1)),Circle(B,sqrt(960),(0,0,1)))[0];
 +
OA = (0,0,-4);
 +
OB = (0,sqrt(560),-8);
 +
OC = (C.x,C.y,-16);
 +
 
 +
draw(shift(OA)*scale3(11)*unitsphere,yellow,light=White);
 +
draw(shift(OB)*scale3(13)*unitsphere,yellow,light=White);
 +
draw(shift(OC)*scale3(19)*unitsphere,yellow,light=White);
 +
draw(Circle(A,sqrt(105),(0,0,1))^^Circle(B,sqrt(105),(0,0,1))^^Circle(C,sqrt(105),(0,0,1)));
 +
draw((-35,-15,0)--(-35,40,0)--(50,40,0)--(50,-15,0)--cycle);
 +
 
 +
dot(OA^^OB^^OC,linewidth(4.5));
 +
dot("$A$",A,(0,1,0),linewidth(4.5));
 +
dot("$B$",B,(0,1,0),linewidth(4.5));
 +
dot("$C$",C,(0,1.5,0),linewidth(4.5));
 +
</asy>
 +
~MRENTHUSIASM
 +
 
 +
==Solution 1==
  
 +
We let <math>\ell</math> be the plane that passes through the spheres and <math>O_A</math> and <math>O_B</math> be the centers of the spheres with radii <math>11</math> and <math>13</math>. We take a cross-section that contains <math>A</math> and <math>B</math>, which contains these two spheres but not the third, as shown below:
 
<asy>
 
<asy>
−
 
+
size(400);
−
size(500);
 
 
pair A, B, OA, OB;
 
pair A, B, OA, OB;
  
Line 18: Line 47:
  
 
draw((-48,0)--(24,0));
 
draw((-48,0)--(24,0));
−
label("$l$",(-42,1),N);
+
label("$\ell$",(-42,1),N);
  
 
label("$A$",A,N);
 
label("$A$",A,N);
Line 41: Line 70:
 
label("$x$",midpoint(midpoint(B--OB)--OB),E);
 
label("$x$",midpoint(midpoint(B--OB)--OB),E);
 
label("$D$",midpoint(B--OB),E);
 
label("$D$",midpoint(B--OB),E);
 +
</asy>
 +
Because the plane cuts out congruent circles, they have the same radius and from the given information, <math>AB = \sqrt{560}</math>. Since <math>ABO_BO_A</math> is a trapezoid, we can drop an altitude from <math>O_A</math> to <math>BO_B</math> to create a rectangle and triangle to use Pythagorean theorem. We know that the length of the altitude is <math>\sqrt{560}</math> and let the distance from <math>O_B</math> to <math>D</math> be <math>x</math>. Then we have <math>x^2 = 576-560 \implies x = 4</math>.
  
 +
We have <math>AO_A = BD</math> because of the rectangle, so <math>\sqrt{11^2-r^2} = \sqrt{13^2-r^2}-4</math>.
 +
Squaring, we have <math>121-r^2 = 169-r^2 + 16 - 8 \cdot \sqrt{169-r^2}</math>.
 +
Subtracting, we get <math>8 \cdot \sqrt{169-r^2} = 64 \implies \sqrt{169-r^2} = 8 \implies 169-r^2 = 64 \implies r^2 = 105</math>.
 +
We also notice that since we had <math>\sqrt{169-r^2} = 8</math> means that <math>BO_B = 8</math> and since we know that <math>x = 4</math>, <math>AO_A = 4</math>.
  
−
 
+
We take a cross-section that contains <math>A</math> and <math>C</math>, which contains these two spheres but not the third, as shown below:
−
</asy>
 
−
 
 
 
<asy>
 
<asy>
−
 
+
size(400);
−
size(500);
 
 
pair A, C, OA, OC;
 
pair A, C, OA, OC;
  
Line 60: Line 92:
  
 
draw((-48,0)--(24,0));
 
draw((-48,0)--(24,0));
−
label("$l$",(-42,1),N);
+
label("$\ell$",(-42,1),N);
  
 
label("$A$",A,N);
 
label("$A$",A,N);
Line 81: Line 113:
 
label("$r$",midpoint(C--(10.2469508,0)),N);
 
label("$r$",midpoint(C--(10.2469508,0)),N);
 
label("$E$",(0,-4),E);
 
label("$E$",(0,-4),E);
−
 
−
 
−
 
 
</asy>
 
</asy>
−
 
+
We have <math>CO_C = \sqrt{19^2-r^2} = \sqrt{361 - 105} = \sqrt{256} = 16</math>. Since <math>AO_A = 4</math>, we have <math>EO_C = 16-4 = 12</math>. Using Pythagorean theorem, <math>O_AE = \sqrt{30^2 - 12^2} = \sqrt{900-144} = \sqrt{756}</math>. Therefore, <math>O_AE^2 = AC^2  = \boxed{756}</math>.
−
==Solution 1==
 
−
 
 
−
We let <math>l</math> be the plane that passes through the spheres and <math>O_A</math> and <math>O_B</math> be the centers of the spheres with radii <math>11</math> and <math>13</math>. We take a cross-section that contains <math>A</math> and <math>B</math>, which contains these two spheres but not the third. Because the plane cuts out congruent circles, they have the same radius and from the given information, <math>AB = \sqrt{560}</math>. Since <math>ABO_BO_A</math> is a trapezoid, we can drop an altitude from <math>O_A</math> to <math>BO_B</math> to create a rectangle and triangle to use Pythagorean theorem. We know that the length of the altitude is <math>\sqrt{560}</math> and let the distance from <math>O_B</math> to <math>D</math> be <math>x</math>. Then we have <math>x^2 = 576-560 \implies x = 4</math>.
 
−
 
 
−
We have <math>AO_A = BD</math> because of the rectangle, so <math>\sqrt{11^2-r^2} = \sqrt{13^2-r^2}-4</math>.
 
−
Squaring, we have <math>121-r^2 = 169-r^2 + 16 - 8 \cdot \sqrt{169-r^2}</math>.
 
−
Subtracting, we get <math>8 \cdot \sqrt{169-r^2} = 64 \implies \sqrt{169-r^2} = 8 \implies 169-r^2 = 64 \implies r^2 = 105</math>.
 
−
We also notice that since we had <math>\sqrt{169-r^2} = 8</math> means that <math>BO_B = 8</math> and since we know that <math>x = 4</math>, <math>AO_A = 4</math>.
 
−
 
 
−
We now look at our second diagram.
 
−
 
 
−
<math>CO_C = \sqrt{19^2-r^2} = \sqrt{361 - 105} = \sqrt{256} = 16</math>. Since <math>AO_A = 4</math>, we have <math>EO_C = 16-4 = 12</math>. Using Pythagorean theorem, <math>O_AE = \sqrt{30^2 - 12^2} = \sqrt{900-144} = \sqrt{756}</math>. Therefore, <math>O_AE^2 = AC^2  = \boxed{756}</math>
 
  
 
~KingRavi
 
~KingRavi

Revision as of 15:35, 12 January 2023

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 $AC^2$.

Diagram

[asy] /* Made by MRENTHUSIASM */ size(250); import graph3; import solids; currentprojection=orthographic((7,0.2,2)); triple A, B, C, OA, OB, OC; A = (0,0,0); B = (0,sqrt(560),0); C = intersectionpoints(Circle(A,sqrt(756),(0,0,1)),Circle(B,sqrt(960),(0,0,1)))[0]; OA = (0,0,-4); OB = (0,sqrt(560),-8); OC = (C.x,C.y,-16); draw(shift(OA)*scale3(11)*unitsphere,yellow,light=White); draw(shift(OB)*scale3(13)*unitsphere,yellow,light=White); draw(shift(OC)*scale3(19)*unitsphere,yellow,light=White); draw(Circle(A,sqrt(105),(0,0,1))^^Circle(B,sqrt(105),(0,0,1))^^Circle(C,sqrt(105),(0,0,1))); draw((-35,-15,0)--(-35,40,0)--(50,40,0)--(50,-15,0)--cycle); dot(OA^^OB^^OC,linewidth(4.5)); dot("$A$",A,(0,1,0),linewidth(4.5)); dot("$B$",B,(0,1,0),linewidth(4.5)); dot("$C$",C,(0,1.5,0),linewidth(4.5)); [/asy] ~MRENTHUSIASM

Solution 1

We let $\ell$ be the plane that passes through the spheres and $O_A$ and $O_B$ be the centers of the spheres with radii $11$ and $13$. We take a cross-section that contains $A$ and $B$, which contains these two spheres but not the third, as shown below: [asy] size(400); 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((-48,0)--(24,0)); label("$\ell$",(-42,1),N); label("$A$",A,N); label("$B$",B,N); label("$O_A$",OA,S); label("$O_B$",OB,S); draw(A--OA); draw(B--OB); draw(OA--OB); draw(OA--(0,-4)); draw(OA--(-33.9112699,0)); draw(OB--(10.2469508,0)); label("$24$",midpoint(OA--OB),S); label("$\sqrt{560}$",midpoint(A--B),N); label("$11$",midpoint(OA--(-33.9112699,0)),S); label("$13$",midpoint(OB--(10.2469508,0)),S); label("$r$",midpoint(midpoint(A--B)--A),N); label("$r$",midpoint(midpoint(A--B)--B),N); label("$r$",midpoint(A--(-33.9112699,0)),N); label("$r$",midpoint(B--(10.2469508,0)),N); label("$x$",midpoint(midpoint(B--OB)--OB),E); label("$D$",midpoint(B--OB),E); [/asy] Because the plane cuts out congruent circles, they have the same radius and from the given information, $AB = \sqrt{560}$. Since $ABO_BO_A$ is a trapezoid, we can drop an altitude from $O_A$ to $BO_B$ to create a rectangle and triangle to use Pythagorean theorem. We know that the length of the altitude is $\sqrt{560}$ and let the distance from $O_B$ to $D$ be $x$. Then we have $x^2 = 576-560 \implies x = 4$.

We have $AO_A = BD$ because of the rectangle, so $\sqrt{11^2-r^2} = \sqrt{13^2-r^2}-4$. Squaring, we have $121-r^2 = 169-r^2 + 16 - 8 \cdot \sqrt{169-r^2}$. Subtracting, we get $8 \cdot \sqrt{169-r^2} = 64 \implies \sqrt{169-r^2} = 8 \implies 169-r^2 = 64 \implies r^2 = 105$. We also notice that since we had $\sqrt{169-r^2} = 8$ means that $BO_B = 8$ and since we know that $x = 4$, $AO_A = 4$.

We take a cross-section that contains $A$ and $C$, which contains these two spheres but not the third, as shown below: [asy] size(400); pair A, C, OA, OC; C = (0,0); A = (-27.4954541697,0); OC = (0,-16); OA = (-27.4954541697,-4); draw(circle(OC,19)); draw(circle(OA,11)); draw((-48,0)--(24,0)); label("$\ell$",(-42,1),N); label("$A$",A,N); label("$C$",C,N); label("$O_A$",OA,S); label("$O_C$",OC,S); draw(A--OA); draw(C--OC); draw(OA--OC); draw(OA--(0,-4)); draw(OA--(-37.8877590151,0)); draw(OC--(10.2469508,0)); label("$30$",midpoint(OA--OC),S); label("$11$",midpoint(OA--(-37.8877590151,0)),S); label("$19$",midpoint(OC--(10.2469508,0)),E); label("$r$",midpoint(midpoint(A--C)--A),N); label("$r$",midpoint(midpoint(A--C)--C),N); label("$r$",midpoint(A--(-37.8877590151,0)),N); label("$r$",midpoint(C--(10.2469508,0)),N); label("$E$",(0,-4),E); [/asy] We have $CO_C = \sqrt{19^2-r^2} = \sqrt{361 - 105} = \sqrt{256} = 16$. Since $AO_A = 4$, we have $EO_C = 16-4 = 12$. Using Pythagorean theorem, $O_AE = \sqrt{30^2 - 12^2} = \sqrt{900-144} = \sqrt{756}$. Therefore, $O_AE^2 = AC^2 = \boxed{756}$.

~KingRavi

Solution 2

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 3

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, $O_A O_B = 11 + 13 = 24$, $O_A O_C = 11 + 19 = 30$.

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 (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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