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Difference between revisions of "2021 AIME II Problems/Problem 10"

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<u><b>Remarks</b></u>
 
<u><b>Remarks</b></u>
 
<ol style="margin-left: 1.5em;">
 
<ol style="margin-left: 1.5em;">
−
   <li>Let <math>\mathcal{R}</math> be the plane that is determined by the centers of the spheres, as shown in the black points. Clearly, the side-lengths of the black dashed triangle are <math>49,49,</math> and <math>72.</math> Note that the four black dashed line segments all lie in plane <math>\mathcal{R}.</math></li><p>
+
   <li>Let <math>\mathcal{R}</math> be the plane that is determined by the centers of the spheres, as shown in the black points. Clearly, the side-lengths of the black dashed triangle are <math>49,49,</math> and <math>72.</math></li><p>
−
   <li>Plane <math>\mathcal{P}</math> is tangent to the spheres at the green points. Therefore, the blue dashed line segments are the radii of the spheres. Note that the four green solid line segments all lie in plane <math>\mathcal{P}.</math></li><p>
+
   <li>Plane <math>\mathcal{P}</math> is tangent to the spheres at the green points. Therefore, the blue dashed line segments are the radii of the spheres.</li><p>
−
  <li>By symmetry, since planes <math>\mathcal{P}</math> and <math>\mathcal{Q}</math> are reflections of each other about plane <math>\mathcal{R},</math> it follows that the three planes are concurrent to line <math>\ell.</math> So, the red point (the foot of the perpendicular from the smallest sphere's center to line <math>\ell</math>) lies in all three planes.</li><p>
+
  <li>We can conclude all of the following:
 +
      <ul style="list-style-type:square; margin-left: 1.5em;">
 +
        <li>The four black dashed line segments all lie in plane <math>\mathcal{R}.</math></li><p>
 +
        <li>The four green solid line segments all lie in plane <math>\mathcal{P}.</math></li><p>
 +
        <li>By symmetry, since planes <math>\mathcal{P}</math> and <math>\mathcal{Q}</math> are reflections of each other about plane <math>\mathcal{R},</math> the three planes are concurrent to line <math>\ell.</math></li><p>
 +
        <li>The red point is the foot of the perpendicular from the smallest sphere's center to line <math>\ell.</math></li><p>
 +
      </ul>
 +
</li><p>
 
</ol>
 
</ol>
  
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==Solution 1==
 
==Solution 1==
  
−
The centers of the three spheres form a 49-49-72 triangle. Consider the points at which the plane is tangent to the two bigger spheres; the line segment connecting these two points should be parallel to the 72 side of this triangle. Take its midpoint <math>M</math>, which is 36 away from the midpoint of the 72 side <math>A</math>, and connect these two midpoints.
+
The centers of the three spheres form a <math>49</math>-<math>49</math>-<math>72</math> triangle. Consider the points at which the plane is tangent to the two bigger spheres; the line segment connecting these two points should be parallel to the <math>72</math> side of this triangle. Take its midpoint <math>M</math>, which is <math>36</math> away from the midpoint <math>A</math> of the <math>72</math> side, and connect these two midpoints.
  
−
Now consider the point at which the plane is tangent to the small sphere, and connect <math>M</math> with the small sphere's tangent point <math>B</math>. Extend <math>MB</math> through B until it hits the ray from <math>A</math> through the center of the small sphere (convince yourself that these two intersect). Call this intersection <math>D</math>, the center of the small sphere <math>C</math>, we want to find <math>BD</math>.
+
Now consider the point at which the plane is tangent to the small sphere, and connect <math>M</math> with the small sphere's tangent point <math>B</math>. Extend <math>\overline{MB}</math> through <math>B</math> until it hits the ray from <math>A</math> through the center of the small sphere (convince yourself that these two intersect). Call this intersection <math>D</math>, the center of the small sphere <math>C</math>, we want to find <math>BD</math>.
  
−
By Pythagorus AC= <math>\sqrt{49^2-36^2}=\sqrt{1105}</math>, and we know <math>MB=36,BC=13</math>. We know that <math>MB,BC</math> must be parallel, using ratios we realize that <math>CD=\frac{13}{23}\sqrt{1105}</math>. Apply Pythagorean theorem on triangle BCD; <math>BD=\frac{312}{23}</math>, so 312 + 23 = <math>\boxed{335}</math>
+
By Pythagorus, <math>AC=\sqrt{49^2-36^2}=\sqrt{1105}</math>, and we know that <math>MB=36</math> and <math>BC=13</math>. We know that <math>\overline{MB}</math> and <math>\overline{BC}</math> must be parallel, using ratios we realize that <math>CD=\frac{13}{23}\sqrt{1105}</math>. Apply the Pythagorean theorem to <math>\triangle BCD</math>, <math>BD=\frac{312}{23}</math>, so <math>312 + 23 = \boxed{335}</math>.
  
 
-Ross Gao
 
-Ross Gao
  
−
==Solution 2 (Coord Bash)==
+
==Solution 2 (Coordinates Bash)==
−
Let's try to see some symmetry. We can use a coordinate plane to plot where the circles are. The 2 large spheres are externally tangent, so we'll make them at 0, -36, 0 and 0, 36, 0. The center of the little sphere would be x, 0, and -23 since we don't know how much the little sphere will be "pushed" down. We use the 3D distance formula to find that x is -24 (since 24 wouldn't make sense). Now, we draw a line through the little sphere and the origin. It also intersects <math>\ell</math> because of the symmetry we created.  
+
Let's try to see some symmetry. We can use an <math>xyz</math>-plane to plot where the circles are. The two large spheres are externally tangent, so we'll make them at <math>(0,-36,0)</math> and <math>(0,36,0)</math>. The center of the little sphere would be <math>(x,0,-23)</math> since we don't know how much the little sphere will be "pushed" down. We use the 3D distance formula to find that <math>x=-24</math> (since <math>x=24</math> wouldn't make sense). Now, we draw a line through the little sphere and the origin. It also intersects <math>\ell</math> because of the symmetry we created.  
  
−
<math>\ell</math> lies on the plane too, so these 2 lines must intersect. The point at where it intersects is -24a, 0, and 23a. We can use the distance formula again to find that a = <math>\dfrac{36}{23}</math>. Therefore, they intersect at <math>\left(-\dfrac{864}{23},0,-36\right)</math>. Since the little circle's x coordinate is -24 and the intersection point's x coordinate is <math>\dfrac{864}{23}</math>, we get <math>\dfrac{864}{23}</math> - 24 = <math>\dfrac{312}{23}</math>. Therefore, our answer to this problem is 312 + 23 =  <math>\boxed{335}</math>.  
+
<math>\ell</math> lies on the plane too, so these two lines must intersect. The point at where it intersects is <math>(-24a,0,23a)</math>. We can use the distance formula again to find that <math>a=\dfrac{36}{23}</math>. Therefore, they intersect at <math>\left(-\dfrac{864}{23},0,-36\right)</math>. Since the little circle's <math>x</math>-coordinate is <math>-24</math> and the intersection point's <math>x</math>-coordinate is <math>\dfrac{864}{23}</math>, we get <math>\dfrac{864}{23} - 24 = \dfrac{312}{23}</math>. Therefore, our answer to this problem is <math>312 + 23 =  \boxed{335}</math>.  
  
 
~Arcticturn
 
~Arcticturn
  
−
==Solution 3 (Illustration of Solution 1)==
+
==Solution 3 (Similar Triangles and Pythagorean Theorem)==
 
This solution refers to the <b>Diagram</b> section.
 
This solution refers to the <b>Diagram</b> section.
  
−
<b>Diagram in progress.</b>
+
As shown below, let <math>O_1,O_2,O_3</math> be the centers of the spheres (where sphere <math>O_3</math> is the smallest) and <math>T_1,T_2,T_3</math> be their respective points of tangency to plane <math>\mathcal{P}.</math> Suppose <math>A</math> is the foot of the perpendicular from <math>O_3</math> to line <math>\ell,</math> so <math>\overleftrightarrow{O_3A}</math> is the perpendicular bisector of <math>\overline{O_1O_2}.</math> We wish to find <math>T_3A.</math>
  
−
As shown above, let <math>O_1,O_2,O_3</math> be the centers of the spheres (where sphere <math>O_3</math> is the smallest) and <math>T_1,T_2,T_3</math> be their respective points of tangency to plane <math>\mathcal{P}.</math> Suppose <math>A</math> is the foot of the perpendicular from <math>O_3</math> to line <math>\ell.</math> We wish to find <math>T_3A.</math>
+
[[File:2021 AIME II Problem 10 Solution 1.png|center]]
  
−
<b>Diagram in progress.</b>
+
As the intersection of planes <math>\mathcal{R}</math> and <math>\mathcal{P}</math> is line <math>\ell,</math> we know that both <math>\overrightarrow{O_1O_3}</math> and <math>\overrightarrow{T_1T_3}</math> must intersect line <math>\ell.</math> Furthermore, since <math>\overline{O_1T_1}\perp\mathcal{P}</math> and <math>\overline{O_3T_3}\perp\mathcal{P},</math> it follows that <math>\overline{O_1T_1}\parallel\overline{O_3T_3},</math> from which <math>O_1,O_3,T_1,</math> and <math>T_3</math> are coplanar.
  
−
<b>Solution in progress. A million thanks for not editing it. I will finish within today.</b>
+
Now, we focus on cross-sections <math>O_1O_3T_3T_1</math> and <math>\mathcal{R}:</math>
 +
<ol style="margin-left: 1.5em;">
 +
  <li><i><b>In the three-dimensional space, the intersection of a line and a plane must be exactly one of the empty set, a point, or a line.</b></i><p>
 +
Clearly, cross-section <math>O_1O_3T_3T_1</math> intersects line <math>\ell</math> at exactly one point. Let the intersection of <math>\overrightarrow{O_1O_3}</math> and line <math>\ell</math> be <math>B,</math> which must also be the intersection of <math>\overrightarrow{T_1T_3}</math> and line <math>\ell.</math></li>
 +
  <li>In cross-section <math>\mathcal{R},</math> let <math>C</math> be the foot of the perpendicular from <math>O_1</math> to line <math>\ell,</math> and <math>D</math> be the foot of the perpendicular from <math>O_3</math> to <math>\overline{O_1C}.</math></li><p>
 +
</ol>
 +
We have the following diagram:
 +
 
 +
[[File:2021 AIME II Problem 10 Solution 2.png|center]]
 +
 
 +
In cross-section <math>O_1O_3T_3T_1,</math> since <math>\overline{O_1T_1}\parallel\overline{O_3T_3}</math> as discussed, we obtain <math>\triangle O_1T_1B\sim\triangle O_3T_3B</math> by AA, with the ratio of similitude <math>\frac{O_1T_1}{O_3T_3}=\frac{36}{13}.</math> Therefore, we get <math>\frac{O_1B}{O_3B}=\frac{49+O_3B}{O_3B}=\frac{36}{13},</math> or <math>O_3B=\frac{637}{23}.</math>
 +
 
 +
In cross-section <math>\mathcal{R},</math> note that <math>O_1O_3=49</math> and <math>DO_3=\frac{O_1O_2}{2}=36.</math> Applying the Pythagorean Theorem to right <math>\triangle O_1DO_3,</math> we have <math>O_1D=\sqrt{1105}.</math> Moreover, since <math>\ell\perp\overline{O_1C}</math> and <math>\overline{DO_3}\perp\overline{O_1C},</math> we obtain <math>\ell\parallel\overline{DO_3}</math> so that <math>\triangle O_1CB\sim\triangle O_1DO_3</math> by AA, with the ratio of similitude <math>\frac{O_1B}{O_1O_3}=\frac{49+\frac{637}{23}}{49}.</math> Therefore, we get <math>\frac{O_1C}{O_1D}=\frac{\sqrt{1105}+DC}{\sqrt{1105}}=\frac{49+\frac{637}{23}}{49},</math> or <math>DC=\frac{13\sqrt{1105}}{23}.</math>
 +
 
 +
Finally, note that <math>\overline{O_3T_3}\perp\overline{T_3A}</math> and <math>O_3T_3=13.</math> Since <math>DCAO_3</math> is a rectangle, we have <math>O_3A=DC=\frac{13\sqrt{1105}}{23}.</math> Applying the Pythagorean Theorem to right <math>\triangle O_3T_3A</math> gives <math>T_3A=\frac{312}{23},</math> from which the answer is <math>312+23=\boxed{335}.</math>
  
 
~MRENTHUSIASM
 
~MRENTHUSIASM

Revision as of 10:53, 20 June 2021

Problem

Two spheres with radii $36$ and one sphere with radius $13$ are each externally tangent to the other two spheres and to two different planes $\mathcal{P}$ and $\mathcal{Q}$. The intersection of planes $\mathcal{P}$ and $\mathcal{Q}$ is the line $\ell$. The distance from line $\ell$ to the point where the sphere with radius $13$ is tangent to plane $\mathcal{P}$ is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Diagram

File:2021 AIME II Problem 10 Diagram.png

Remarks

  1. Let $\mathcal{R}$ be the plane that is determined by the centers of the spheres, as shown in the black points. Clearly, the side-lengths of the black dashed triangle are $49,49,$ and $72.$

  2. Plane $\mathcal{P}$ is tangent to the spheres at the green points. Therefore, the blue dashed line segments are the radii of the spheres.

  3. We can conclude all of the following:
    • The four black dashed line segments all lie in plane $\mathcal{R}.$

    • The four green solid line segments all lie in plane $\mathcal{P}.$

    • By symmetry, since planes $\mathcal{P}$ and $\mathcal{Q}$ are reflections of each other about plane $\mathcal{R},$ the three planes are concurrent to line $\ell.$

    • The red point is the foot of the perpendicular from the smallest sphere's center to line $\ell.$

~MRENTHUSIASM (by Geometry Expressions)

Solution 1

The centers of the three spheres form a $49$-$49$-$72$ triangle. Consider the points at which the plane is tangent to the two bigger spheres; the line segment connecting these two points should be parallel to the $72$ side of this triangle. Take its midpoint $M$, which is $36$ away from the midpoint $A$ of the $72$ side, and connect these two midpoints.

Now consider the point at which the plane is tangent to the small sphere, and connect $M$ with the small sphere's tangent point $B$. Extend $\overline{MB}$ through $B$ until it hits the ray from $A$ through the center of the small sphere (convince yourself that these two intersect). Call this intersection $D$, the center of the small sphere $C$, we want to find $BD$.

By Pythagorus, $AC=\sqrt{49^2-36^2}=\sqrt{1105}$, and we know that $MB=36$ and $BC=13$. We know that $\overline{MB}$ and $\overline{BC}$ must be parallel, using ratios we realize that $CD=\frac{13}{23}\sqrt{1105}$. Apply the Pythagorean theorem to $\triangle BCD$, $BD=\frac{312}{23}$, so $312 + 23 = \boxed{335}$.

-Ross Gao

Solution 2 (Coordinates Bash)

Let's try to see some symmetry. We can use an $xyz$-plane to plot where the circles are. The two large spheres are externally tangent, so we'll make them at $(0,-36,0)$ and $(0,36,0)$. The center of the little sphere would be $(x,0,-23)$ since we don't know how much the little sphere will be "pushed" down. We use the 3D distance formula to find that $x=-24$ (since $x=24$ wouldn't make sense). Now, we draw a line through the little sphere and the origin. It also intersects $\ell$ because of the symmetry we created.

$\ell$ lies on the plane too, so these two lines must intersect. The point at where it intersects is $(-24a,0,23a)$. We can use the distance formula again to find that $a=\dfrac{36}{23}$. Therefore, they intersect at $\left(-\dfrac{864}{23},0,-36\right)$. Since the little circle's $x$-coordinate is $-24$ and the intersection point's $x$-coordinate is $\dfrac{864}{23}$, we get $\dfrac{864}{23} - 24 = \dfrac{312}{23}$. Therefore, our answer to this problem is $312 + 23 = \boxed{335}$.

~Arcticturn

Solution 3 (Similar Triangles and Pythagorean Theorem)

This solution refers to the Diagram section.

As shown below, let $O_1,O_2,O_3$ be the centers of the spheres (where sphere $O_3$ is the smallest) and $T_1,T_2,T_3$ be their respective points of tangency to plane $\mathcal{P}.$ Suppose $A$ is the foot of the perpendicular from $O_3$ to line $\ell,$ so $\overleftrightarrow{O_3A}$ is the perpendicular bisector of $\overline{O_1O_2}.$ We wish to find $T_3A.$

File:2021 AIME II Problem 10 Solution 1.png

As the intersection of planes $\mathcal{R}$ and $\mathcal{P}$ is line $\ell,$ we know that both $\overrightarrow{O_1O_3}$ and $\overrightarrow{T_1T_3}$ must intersect line $\ell.$ Furthermore, since $\overline{O_1T_1}\perp\mathcal{P}$ and $\overline{O_3T_3}\perp\mathcal{P},$ it follows that $\overline{O_1T_1}\parallel\overline{O_3T_3},$ from which $O_1,O_3,T_1,$ and $T_3$ are coplanar.

Now, we focus on cross-sections $O_1O_3T_3T_1$ and $\mathcal{R}:$

  1. In the three-dimensional space, the intersection of a line and a plane must be exactly one of the empty set, a point, or a line.

    Clearly, cross-section $O_1O_3T_3T_1$ intersects line $\ell$ at exactly one point. Let the intersection of $\overrightarrow{O_1O_3}$ and line $\ell$ be $B,$ which must also be the intersection of $\overrightarrow{T_1T_3}$ and line $\ell.$

  2. In cross-section $\mathcal{R},$ let $C$ be the foot of the perpendicular from $O_1$ to line $\ell,$ and $D$ be the foot of the perpendicular from $O_3$ to $\overline{O_1C}.$

We have the following diagram:

File:2021 AIME II Problem 10 Solution 2.png

In cross-section $O_1O_3T_3T_1,$ since $\overline{O_1T_1}\parallel\overline{O_3T_3}$ as discussed, we obtain $\triangle O_1T_1B\sim\triangle O_3T_3B$ by AA, with the ratio of similitude $\frac{O_1T_1}{O_3T_3}=\frac{36}{13}.$ Therefore, we get $\frac{O_1B}{O_3B}=\frac{49+O_3B}{O_3B}=\frac{36}{13},$ or $O_3B=\frac{637}{23}.$

In cross-section $\mathcal{R},$ note that $O_1O_3=49$ and $DO_3=\frac{O_1O_2}{2}=36.$ Applying the Pythagorean Theorem to right $\triangle O_1DO_3,$ we have $O_1D=\sqrt{1105}.$ Moreover, since $\ell\perp\overline{O_1C}$ and $\overline{DO_3}\perp\overline{O_1C},$ we obtain $\ell\parallel\overline{DO_3}$ so that $\triangle O_1CB\sim\triangle O_1DO_3$ by AA, with the ratio of similitude $\frac{O_1B}{O_1O_3}=\frac{49+\frac{637}{23}}{49}.$ Therefore, we get $\frac{O_1C}{O_1D}=\frac{\sqrt{1105}+DC}{\sqrt{1105}}=\frac{49+\frac{637}{23}}{49},$ or $DC=\frac{13\sqrt{1105}}{23}.$

Finally, note that $\overline{O_3T_3}\perp\overline{T_3A}$ and $O_3T_3=13.$ Since $DCAO_3$ is a rectangle, we have $O_3A=DC=\frac{13\sqrt{1105}}{23}.$ Applying the Pythagorean Theorem to right $\triangle O_3T_3A$ gives $T_3A=\frac{312}{23},$ from which the answer is $312+23=\boxed{335}.$

~MRENTHUSIASM

See Also

2021 AIME II (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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