Difference between revisions of "1990 AIME Problems/Problem 12"

Latest revision as of 12:23, 5 September 2021

Problem

A regular 12-gon is inscribed in a circle of radius 12. The sum of the lengths of all sides and diagonals of the 12-gon can be written in the form $a + b \sqrt{2} + c \sqrt{3} + d \sqrt{6},$ where $a^{}_{}$ , $b^{}_{}$ , $c^{}_{}$ , and $d^{}_{}$ are positive integers. Find $a + b + c + d^{}_{}$ .

Solution 1

The easiest way to do this seems to be to find the length of each of the sides and diagonals. To do such, draw the radii that meet the endpoints of the sides/diagonals; this will form isosceles triangles. Drawing the altitude of those triangles and then solving will yield the respective lengths.

The length of each of the 12 sides is $2 \cdot 12\sin 15$ . $24\sin 15 = 24\sin (45 - 30) = 24\frac{\sqrt{6} - \sqrt{2}}{4} = 6(\sqrt{6} - \sqrt{2})$ .
The length of each of the 12 diagonals that span across 2 edges is $2 \cdot 12\sin 30 = 12$ (or notice that the triangle formed is equilateral).
The length of each of the 12 diagonals that span across 3 edges is $2 \cdot 12\sin 45 = 12\sqrt{2}$ (or notice that the triangle formed is a $45 - 45 - 90$ right triangle).
The length of each of the 12 diagonals that span across 4 edges is $2 \cdot 12\sin 60 = 12\sqrt{3}$ .
The length of each of the 12 diagonals that span across 5 edges is $2 \cdot 12\sin 75 = 24\sin (45 + 30) = 24\frac{\sqrt{6}+\sqrt{2}}{4} = 6(\sqrt{6}+\sqrt{2})$ .
The length of each of the 6 diameters is $2 \cdot 12 = 24$ .

Adding all of these up, we get $12[6(\sqrt{6} - \sqrt{2}) + 12 + 12\sqrt{2} + 12\sqrt{3} + 6(\sqrt{6}+\sqrt{2})] + 6 \cdot 24$

$= 12(12 + 12\sqrt{2} + 12\sqrt{3} + 12\sqrt{6}) + 144 = 288 + 144\sqrt{2} + 144\sqrt{3} + 144\sqrt{6}$ . Thus, the answer is $144 \cdot 5 = \boxed{720}$ .

Solution 2

A second method involves drawing a triangle connecting the center of the 12-gon to two vertices of the 12-gon. Since the distance from the center to a vertex of the 12-gon is $12$ , the Law of Cosines can be applied to this isosceles triangle, to give:

$a^2 = 12^2 + 12^2 - 2\cdot 12\cdot 12\cdot \cos \theta$

$a^2 = 2\cdot 12^2 - 2\cdot 12^2 \cos \theta$

$a^2 = 2\cdot 12^2 (1 - \cos \theta)$

$a = 12\sqrt{2} \cdot \sqrt{1 - \cos \theta}$

There are six lengths of sides/diagonals, corresponding to $\theta = {30^{\circ}, 60^{\circ}, 90^{\circ}, 120^{\circ}, 150^{\circ}, 180^{\circ}}$

Call these lengths $a_1, a_2, a_3, a_4, a_5, a_6$ from shortest to longest. The total length $l$ that is asked for is

$l = 12(a_1 + a_2 + a_3 + a_4 + a_5) + 6a_6$ , noting that $a_6$ as written gives the diameter of the circle, which is the longest diagonal.

$l = 12[12\sqrt{2} (\sqrt{1 - \cos 30^{\circ}}+\sqrt{1-\cos 60^{\circ}}+\sqrt{1-\cos 90^{\circ}}+\sqrt{1-\cos 120^{\circ}}+\sqrt{1 - \cos 150^{\circ}})] + 6 \cdot 24$

$l = 144\sqrt{2} \left(\sqrt{1 - \frac{\sqrt{3}}{2}} + \sqrt{1 - \frac{1}{2}} + \sqrt{1-0} + \sqrt{1 + \frac{1}{2}} + \sqrt{1 + \frac{\sqrt{3}}{2}}\right) + 144$

$l = 144\sqrt{2} \left(\sqrt{1 - \frac{\sqrt{3}}{2}} + \frac{\sqrt{2}}{2} + 1 + \frac{\sqrt{6}}{2} + \sqrt{1 + \frac{\sqrt{3}}{2}}\right) + 144$

To simplify the two nested radicals, add them, and call the sum $x$ :

$x = \sqrt{1 - \frac{\sqrt{3}}{2}} + \sqrt{1 + \frac{\sqrt{3}}{2}}$

Squaring both sides, the F and L part of FOIL causes the radicals to cancel, leaving:

$x^2 = 2 + 2\sqrt{\left(1 - \frac{\sqrt{3}}{2}\right)\left(1 + \frac{\sqrt{3}}{2}\right)}$

$x^2 = 2 + 2\sqrt{1 - \frac{3}{4}}$

$x^2 = 2 + 2\sqrt{\frac{1}{4}}$

$x^2 = 2 + 2\cdot\frac{1}{2}$

$x = \sqrt{3}.$

Plugging that sum $x$ back into the equation for $l$ , we find

$l = 144\sqrt{2} \left(\frac{\sqrt{2}}{2} + 1 + \frac{\sqrt{6}}{2} + \sqrt{3}\right) + 144$

$l = 144 + 144\sqrt{2} + 144\sqrt{3} + 144\sqrt{6} + 144$ .

Thus, the desired quantity is $144\cdot 5 = \boxed{720}$ .

Solution 3

Begin as in solution 2, drawing a triangle connecting the center of the 12-gon to two vertices of the 12-gon. Apply law of cosines on $\theta = {30^{\circ}, 60^{\circ}, 90^{\circ}, 120^{\circ}, 150^{\circ}, 180^{\circ}}$ to get $d^2 = 288 - 288 \cos \theta$ where $d$ is the diagonal or sidelength distance between two points on the 12-gon. Now, $d=\sqrt{288-288 \cos \theta}$ . Instead of factoring out $288$ as in solution 2, factor out $\sqrt{576}$ instead-the motivation for this is to make the expression look like the half angle identity, and the fact that $\sqrt{576}$ is an integer doesn't hurt. Now, we have that $d=24 \sin \frac{\theta}{2}$ , which simplifies things quite nicely. Continue as in solution 2, computing half-angle sines instead of nested radicals, to obtain $\boxed{720}$ .

Solution 4 (Another Way to Start Solution 1)

Plotting this regular $12$ -gon on the complex plane with center as origin, and a vertex on the $x$ -axis. We have that the vertices are the of the form $12e^{\frac{2\pi k}{12} i}.$ All of the distinct diagonals can be represented with one vertex as $12e^{\frac{2\pi 0}{12} i} = 12.$ Then the each diagonal can be expressed as, $\begin{align*} 12 \sqrt{(\cos{0} - \cos{\frac{2\pi k}{12}})^2 + (\sin{\frac{2\pi k}{12}} - \sin{0})^2} &= 12 \sqrt{(\cos{0})^2 + (\cos{\frac{2\pi k}{12}})^2 + (\sin{\frac{2\pi k}{12}})^2 + (\sin{0})^2 - 2(\cos{\frac{2\pi k}{12}}\cos{0} + \sin{\frac{2\pi k}{12}}\sin{0})} \\ &= 12 \sqrt{2 - 2\cos{\frac{2\pi k}{12}}} \\ &= 12 \sqrt{4\sin^2{\frac{\pi k}{12}}} \\ &= 24\sin{\frac{\pi k}{12}}.\end{align*}$ The rest follows as Solution 1.

~Aaryabhatta1

@@ Line 2: / Line 2: @@
 A [[regular polygon|regular]] 12-gon is inscribed in a [[circle]] of [[radius]] 12.  The [[sum]] of the lengths of all sides and [[diagonal]]s of the 12-gon can be written in the form <math>a + b \sqrt{2} + c \sqrt{3} + d \sqrt{6},</math> where <math>a^{}_{}</math>, <math>b^{}_{}</math>, <math>c^{}_{}</math>, and <math>d^{}_{}</math> are positive integers.  Find <math>a + b + c + d^{}_{}</math>.
-== Solution ==
+== Solution 1==
-{{image}}
+[[Image:1990 AIME-12.png]]
 The easiest way to do this seems to be to find the length of each of the sides and diagonals. To do such, draw the [[radius|radii]] that meet the endpoints of the sides/diagonals; this will form [[isosceles triangle]]s. Drawing the [[altitude]] of those [[triangle]]s and then solving will yield the respective lengths.
@@ Line 11: / Line 12: @@
 *The length of each of the 12 diagonals that span across 4 edges is <math>2 \cdot 12\sin 60 = 12\sqrt{3}</math>.
 *The length of each of the 12 diagonals that span across 5 edges is <math>2 \cdot 12\sin 75 = 24\sin (45 + 30) = 24\frac{\sqrt{6}+\sqrt{2}}{4} = 6(\sqrt{6}+\sqrt{2})</math>.
-*The length of each of the 6 [[diameter]]s are <math>2 \cdot 12 = 24</math>.
+*The length of each of the 6 [[diameter]]s is <math>2 \cdot 12 = 24</math>.
+Adding all of these up, we get <math>12[6(\sqrt{6} - \sqrt{2}) + 12 + 12\sqrt{2} + 12\sqrt{3} + 6(\sqrt{6}+\sqrt{2})] + 6 \cdot 24</math>
+<math>= 12(12 + 12\sqrt{2} + 12\sqrt{3} + 12\sqrt{6}) + 144 = 288 + 144\sqrt{2} + 144\sqrt{3} + 144\sqrt{6}</math>. Thus, the answer is <math>144 \cdot 5 = \boxed{720}</math>.
+==Solution 2==
+A second method involves drawing a triangle connecting the center of the 12-gon to two vertices of the 12-gon.  Since the distance from the center to a vertex of the 12-gon is <math>12</math>, the Law of Cosines can be applied to this isosceles triangle, to give:
+<math>a^2 = 12^2 + 12^2 - 2\cdot 12\cdot 12\cdot \cos \theta</math>
+<math>a^2 = 2\cdot 12^2 - 2\cdot 12^2 \cos \theta</math>
+<math>a^2 = 2\cdot 12^2 (1 -  \cos \theta)</math>
+<math>a = 12\sqrt{2} \cdot \sqrt{1 - \cos \theta}</math>
+There are six lengths of sides/diagonals, corresponding to <math>\theta = {30^{\circ}, 60^{\circ}, 90^{\circ}, 120^{\circ}, 150^{\circ}, 180^{\circ}}</math>
+Call these lengths <math>a_1, a_2, a_3, a_4, a_5, a_6</math> from shortest to longest.  The total length <math>l</math> that is asked for is
+<math>l = 12(a_1 + a_2 + a_3 + a_4 + a_5) + 6a_6</math>, noting that <math>a_6</math> as written gives the diameter of the circle, which is the longest diagonal.
+<math>l = 12[12\sqrt{2} (\sqrt{1 - \cos 30^{\circ}}+\sqrt{1-\cos 60^{\circ}}+\sqrt{1-\cos 90^{\circ}}+\sqrt{1-\cos 120^{\circ}}+\sqrt{1 - \cos 150^{\circ}})] + 6 \cdot 24</math>
+<math>l = 144\sqrt{2} \left(\sqrt{1 - \frac{\sqrt{3}}{2}} + \sqrt{1 - \frac{1}{2}} + \sqrt{1-0} + \sqrt{1 + \frac{1}{2}} + \sqrt{1 + \frac{\sqrt{3}}{2}}\right) + 144</math>
+<math>l = 144\sqrt{2} \left(\sqrt{1 - \frac{\sqrt{3}}{2}} + \frac{\sqrt{2}}{2} + 1 + \frac{\sqrt{6}}{2} + \sqrt{1 + \frac{\sqrt{3}}{2}}\right) + 144</math>
+To simplify the two nested radicals, add them, and call the sum <math>x</math>:
+<math>x = \sqrt{1 - \frac{\sqrt{3}}{2}} + \sqrt{1 + \frac{\sqrt{3}}{2}}</math>
+Squaring both sides, the F and L part of FOIL causes the radicals to cancel, leaving:
+<math>x^2 = 2 + 2\sqrt{\left(1 - \frac{\sqrt{3}}{2}\right)\left(1 + \frac{\sqrt{3}}{2}\right)}</math>
+<math>x^2 = 2 + 2\sqrt{1 - \frac{3}{4}}</math>
+<math>x^2 = 2 + 2\sqrt{\frac{1}{4}}</math>
+<math>x^2 = 2 + 2\cdot\frac{1}{2}</math>
+<math>x = \sqrt{3}.</math>
+Plugging that sum <math>x</math> back into the equation for <math>l</math>, we find
+<math>l = 144\sqrt{2} \left(\frac{\sqrt{2}}{2} + 1 + \frac{\sqrt{6}}{2} + \sqrt{3}\right) + 144</math>
+<math>l = 144 + 144\sqrt{2} + 144\sqrt{3} + 144\sqrt{6} + 144</math>.
+Thus, the desired quantity is <math>144\cdot 5 = \boxed{720}</math>.
+==Solution 3==
+Begin as in solution 2, drawing a triangle connecting the center of the 12-gon to two vertices of the 12-gon. Apply law of cosines on <math>\theta = {30^{\circ}, 60^{\circ}, 90^{\circ}, 120^{\circ}, 150^{\circ}, 180^{\circ}}</math> to get <math>d^2 = 288 - 288 \cos \theta</math> where <math>d</math> is the diagonal or sidelength distance between two points on the 12-gon. Now, <math>d=\sqrt{288-288 \cos \theta}</math>. Instead of factoring out <math>288</math> as in solution 2, factor out <math>\sqrt{576}</math> instead-the motivation for this is to make the expression look like the half angle identity, and the fact that <math>\sqrt{576}</math> is an integer doesn't hurt. Now, we have that <math>d=24 \sin \frac{\theta}{2}</math>, which simplifies things quite nicely. Continue as in solution 2, computing half-angle sines instead of nested radicals, to obtain <math>\boxed{720}</math>.
+==Solution 4 (Another Way to Start Solution 1)==
+Plotting this regular <math>12</math>-gon on the complex plane with center as origin, and a vertex on the <math>x</math>-axis. We have that the vertices are the of the form <math>12e^{\frac{2\pi k}{12} i}.</math> All of the distinct diagonals can be represented with one vertex as <math>12e^{\frac{2\pi 0}{12} i} = 12.</math> Then the each diagonal can be expressed as, <cmath>\begin{align*} 12 \sqrt{(\cos{0} - \cos{\frac{2\pi k}{12}})^2 + (\sin{\frac{2\pi k}{12}} - \sin{0})^2} &=  12 \sqrt{(\cos{0})^2 + (\cos{\frac{2\pi k}{12}})^2 + (\sin{\frac{2\pi k}{12}})^2 + (\sin{0})^2 - 2(\cos{\frac{2\pi k}{12}}\cos{0} + \sin{\frac{2\pi k}{12}}\sin{0})} \\ &= 12 \sqrt{2 - 2\cos{\frac{2\pi k}{12}}} \\ &= 12 \sqrt{4\sin^2{\frac{\pi k}{12}}} \\ &= 24\sin{\frac{\pi k}{12}}.\end{align*}</cmath> The rest follows as Solution 1.
-Adding all of these up, we get <math>12[6(\sqrt{6} - \sqrt{2}) + 12 + 12\sqrt{2} + 12\sqrt{3} + 6(\sqrt{6}+\sqrt{2})] + 6 \cdot 24 = 12(12 + 12\sqrt{2} + 12\sqrt{3} + 12\sqrt{6}) + 144 = 288 + 144\sqrt{2} + 144\sqrt{3} + 144\sqrt{6}</math>. Thus, the answer is <math>144 \cdot 5 = 720</math>.
+~Aaryabhatta1
 == See also ==
@@ Line 19: / Line 85: @@
 [[Category:Intermediate Geometry Problems]]
+{{MAA Notice}}

1990 AIME (Problems • Answer Key • Resources)
Preceded by Problem 11	Followed by Problem 13
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