Difference between revisions of "2020 AIME I Problems/Problem 6"
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== Problem == | == Problem == | ||
− | A flat board has a circular hole with radius <math>1</math> and a circular hole with radius <math>2</math> such that the distance between the centers of the two holes | + | A flat board has a circular hole with radius <math>1</math> and a circular hole with radius <math>2</math> such that the distance between the centers of the two holes is <math>7.</math> Two spheres with equal radii sit in the two holes such that the spheres are tangent to each other. The square of the radius of the spheres is <math>\tfrac{m}{n},</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n.</math> |
− | == | + | == Diagram == |
+ | <asy> | ||
+ | /* Made by MRENTHUSIASM */ | ||
+ | size(300); | ||
+ | import graph3; | ||
+ | import solids; | ||
+ | currentprojection=orthographic((1,-5,1)); | ||
+ | real r = sqrt(160/13); | ||
− | + | triple C1, C2, B1, B2; | |
+ | C1 = (0,0,sqrt(r^2-1^2)); | ||
+ | C2 = (7,0,sqrt(r^2-2^2)); | ||
+ | B1 = (0,0,0); | ||
+ | B2 = (7,0,0); | ||
+ | draw((-5,-10,0)--(-5,10,0)--(12,10,0)--(12,-10,0)--cycle); | ||
+ | draw(shift(C1)*scale3(r)*unitsphere,yellow,light=White); | ||
+ | draw(shift(C2)*scale3(r)*unitsphere,yellow,light=White); | ||
+ | draw(Circle(B1,1,(0,0,1))); | ||
+ | draw(Circle(B2,2,(0,0,1))); | ||
− | + | dot(C1^^C2^^B1^^B2,linewidth(4.5)); | |
+ | </asy> | ||
+ | ~MRENTHUSIASM | ||
− | -molocyxu | + | == Solution 1 (Pythagorean Theorem) == |
+ | <asy> | ||
+ | size(10cm); | ||
+ | pair A, B, C, D, O, P, H, L, X, Y; | ||
+ | A = (-1, 0); | ||
+ | B = (1, 0); | ||
+ | H = (0, 0); | ||
+ | C = (5, 0); | ||
+ | D = (9, 0); | ||
+ | L = (7, 0); | ||
+ | O = (0, sqrt(160/13 - 1)); | ||
+ | P = (7, sqrt(160/13 - 4)); | ||
+ | X = (0, sqrt(160/13 - 4)); | ||
+ | Y = (O + P) / 2; | ||
+ | |||
+ | draw(A -- O -- B -- cycle); | ||
+ | draw(C -- P -- D -- cycle); | ||
+ | draw(B -- C); | ||
+ | draw(O -- P); | ||
+ | draw(P -- X, dashed); | ||
+ | draw(O -- H, dashed); | ||
+ | draw(P -- L, dashed); | ||
+ | |||
+ | draw(circle(O, sqrt(160/13))); | ||
+ | draw(circle(P, sqrt(160/13))); | ||
+ | path b = brace(L-(0,1), H-(0,1),0.5); | ||
+ | draw(b); | ||
+ | |||
+ | label("$r$", O -- Y, N); | ||
+ | label("$r$", Y -- P, N); | ||
+ | label("$r$", O -- A, NW); | ||
+ | label("$r$", P -- D, NE); | ||
+ | label("$1$", A -- H, N); | ||
+ | label("$2$", L -- D, N); | ||
+ | label("$7$", b, S); | ||
+ | |||
+ | dot(A^^B^^C^^D^^O^^P^^H^^L^^X^^Y,linewidth(4)); | ||
+ | </asy> | ||
+ | Set the common radius to <math>r</math>. First, take the cross section of the sphere sitting in the hole of radius <math>1</math>. If we draw the perpendicular bisector of the chord (the hole) through the circle, this line goes through the center. Connect the center also to where the chord hits the circle, for a right triangle with hypotenuse <math>r</math> and base <math>1</math>. Therefore, the height of this circle outside of the hole is <math>\sqrt{r^2-1}</math>. | ||
+ | |||
+ | The other circle follows similarly for a height (outside the hole) of <math>\sqrt{r^2-4}</math>. Now, if we take the cross section of the entire board, essentially making it two-dimensional, we can connect the centers of the two spheres, then form another right triangle with base <math>7</math>, as given by the problem. The height of this triangle is the difference between the heights of the parts of the two spheres outside the holes, which is <math>\sqrt{r^2-1} - \sqrt{r^2-4}</math>. Now we can set up an equation in terms of <math>r</math> with the Pythagorean theorem: <cmath>\left(\sqrt{r^2-1} - \sqrt{r^2-4}\right)^2 + 7^2 = (2r)^2.</cmath> Simplifying a few times, | ||
+ | <cmath>\begin{align*} | ||
+ | r^2 - 1 - 2\left(\sqrt{(r^2-1)(r^2-4)}\right) + r^2 - 4 + 49 &= 4r^2 \\ | ||
+ | 2r^2-44 &= -2\left(\sqrt{(r^2-1)(r^2-4)}\right) \\ | ||
+ | 22-r^2 &= \left(\sqrt{r^4 - 5r^2 + 4}\right) \\ | ||
+ | r^4 -44r^2 + 484 &= r^4 - 5r^2 + 4 \\ | ||
+ | 39r^2&=480 \\ | ||
+ | r^2&=\frac{480}{39} = \frac{160}{13}. | ||
+ | \end{align*}</cmath> | ||
+ | Therefore, our answer is <math>\boxed{173}</math>. | ||
+ | |||
+ | ~molocyxu | ||
+ | |||
+ | ==Solution 2 (Tangential Distance)== | ||
+ | [[File:2020 AIME I 6a.png|400px|right]] | ||
+ | Let <math>A</math> and <math>B</math> be the centers of the holes, let <math>C</math> be the point of crossing <math>AB</math> and radical axes of the circles. So <math>C</math> has equal tangential distance to any point of both spheres. In particular to the circles (https://en.wikipedia.org/wiki/Radical_axis.) | ||
+ | [[File:2020 AIME I 6d.png|400px|right]] | ||
+ | <cmath>CA = \frac {AB} {2} – \frac {r_A^2 – r_B^2}{2 AB} = \frac{23}{7}, CB = AB - AC =\frac{26}{7},</cmath> | ||
+ | <cmath>CA' = CB'= \sqrt{BC^2 – r_B^2} = \frac {4}{7} \sqrt{30}.</cmath> | ||
+ | |||
+ | Let <math>D</math> be the point of tangency of the spheres common radius <math>R</math> centered at <math>O</math> and <math>O'.</math> Let <math>\alpha</math> be the angle between <math>OO'</math> and flat board. In the plane, perpendicular to board <cmath>DC \perp OO', DC = \frac {4}{7} \sqrt{30}.</cmath> | ||
+ | [[File:2020 AIME I 6b.png|400px|right]] | ||
+ | Distance between <math>C</math> and midpoint <math>M</math> of <math>AB</math> is | ||
+ | <cmath>d = \frac {26}{7} - \frac {7}{2} = \frac{3}{14} \implies \sin \alpha = \frac {d}{DC} = \frac {3}{8\sqrt {30}}.</cmath> | ||
+ | <cmath>\cos \alpha = \sqrt {1 – \frac {9}{64 \cdot 30}} = \sqrt{ \frac {637}{640}} = \frac {7}{2} \sqrt {\frac{13}{160}}.</cmath> | ||
+ | <cmath> 2R \cos \alpha = AB = 7 \implies R = \frac {\frac{7}{2} } {\frac{7}{2}\sqrt \frac{13}{160}} = \sqrt {\frac{160}{13}} .</cmath> | ||
+ | '''vladimir.shelomovskii@gmail.com, vvsss''' | ||
+ | |||
+ | ==Video solution (With Animation)== | ||
+ | |||
+ | https://youtu.be/cOf9uTJ9J40 | ||
+ | |||
+ | ==Video Solution== | ||
+ | |||
+ | https://www.youtube.com/watch?v=qCTq8KhZfYQ | ||
==See Also== | ==See Also== | ||
{{AIME box|year=2020|n=I|num-b=5|num-a=7}} | {{AIME box|year=2020|n=I|num-b=5|num-a=7}} | ||
+ | |||
+ | [[Category: Introductory Geometry Problems]] | ||
+ | [[Category: 3D Geometry Problems]] | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 07:54, 15 December 2023
Contents
Problem
A flat board has a circular hole with radius and a circular hole with radius
such that the distance between the centers of the two holes is
Two spheres with equal radii sit in the two holes such that the spheres are tangent to each other. The square of the radius of the spheres is
where
and
are relatively prime positive integers. Find
Diagram
~MRENTHUSIASM
Solution 1 (Pythagorean Theorem)
Set the common radius to
. First, take the cross section of the sphere sitting in the hole of radius
. If we draw the perpendicular bisector of the chord (the hole) through the circle, this line goes through the center. Connect the center also to where the chord hits the circle, for a right triangle with hypotenuse
and base
. Therefore, the height of this circle outside of the hole is
.
The other circle follows similarly for a height (outside the hole) of . Now, if we take the cross section of the entire board, essentially making it two-dimensional, we can connect the centers of the two spheres, then form another right triangle with base
, as given by the problem. The height of this triangle is the difference between the heights of the parts of the two spheres outside the holes, which is
. Now we can set up an equation in terms of
with the Pythagorean theorem:
Simplifying a few times,
Therefore, our answer is
.
~molocyxu
Solution 2 (Tangential Distance)
Let and
be the centers of the holes, let
be the point of crossing
and radical axes of the circles. So
has equal tangential distance to any point of both spheres. In particular to the circles (https://en.wikipedia.org/wiki/Radical_axis.)
Let be the point of tangency of the spheres common radius
centered at
and
Let
be the angle between
and flat board. In the plane, perpendicular to board
Distance between and midpoint
of
is
vladimir.shelomovskii@gmail.com, vvsss
Video solution (With Animation)
Video Solution
https://www.youtube.com/watch?v=qCTq8KhZfYQ
See Also
2020 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.