Difference between revisions of "2011 AIME I Problems/Problem 14"
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== Problem == | == Problem == | ||
Let <math>A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8</math> be a regular octagon. Let <math>M_1</math>, <math>M_3</math>, <math>M_5</math>, and <math>M_7</math> be the midpoints of sides <math>\overline{A_1 A_2}</math>, <math>\overline{A_3 A_4}</math>, <math>\overline{A_5 A_6}</math>, and <math>\overline{A_7 A_8}</math>, respectively. For <math>i = 1, 3, 5, 7</math>, ray <math>R_i</math> is constructed from <math>M_i</math> towards the interior of the octagon such that <math>R_1 \perp R_3</math>, <math>R_3 \perp R_5</math>, <math>R_5 \perp R_7</math>, and <math>R_7 \perp R_1</math>. Pairs of rays <math>R_1</math> and <math>R_3</math>, <math>R_3</math> and <math>R_5</math>, <math>R_5</math> and <math>R_7</math>, and <math>R_7</math> and <math>R_1</math> meet at <math>B_1</math>, <math>B_3</math>, <math>B_5</math>, <math>B_7</math> respectively. If <math>B_1 B_3 = A_1 A_2</math>, then <math>\cos 2 \angle A_3 M_3 B_1</math> can be written in the form <math>m - \sqrt{n}</math>, where <math>m</math> and <math>n</math> are positive integers. Find <math>m + n</math>. | Let <math>A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8</math> be a regular octagon. Let <math>M_1</math>, <math>M_3</math>, <math>M_5</math>, and <math>M_7</math> be the midpoints of sides <math>\overline{A_1 A_2}</math>, <math>\overline{A_3 A_4}</math>, <math>\overline{A_5 A_6}</math>, and <math>\overline{A_7 A_8}</math>, respectively. For <math>i = 1, 3, 5, 7</math>, ray <math>R_i</math> is constructed from <math>M_i</math> towards the interior of the octagon such that <math>R_1 \perp R_3</math>, <math>R_3 \perp R_5</math>, <math>R_5 \perp R_7</math>, and <math>R_7 \perp R_1</math>. Pairs of rays <math>R_1</math> and <math>R_3</math>, <math>R_3</math> and <math>R_5</math>, <math>R_5</math> and <math>R_7</math>, and <math>R_7</math> and <math>R_1</math> meet at <math>B_1</math>, <math>B_3</math>, <math>B_5</math>, <math>B_7</math> respectively. If <math>B_1 B_3 = A_1 A_2</math>, then <math>\cos 2 \angle A_3 M_3 B_1</math> can be written in the form <math>m - \sqrt{n}</math>, where <math>m</math> and <math>n</math> are positive integers. Find <math>m + n</math>. | ||
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==Diagram== | ==Diagram== | ||
Line 91: | Line 51: | ||
</asy> | </asy> | ||
All distances are to scale. | All distances are to scale. | ||
+ | |||
+ | ==Solution 1== | ||
+ | We use coordinates. Let the octagon have side length <math>2</math> and center <math>(0, 0)</math>. Then all of its vertices have the form <math>(\pm 1, \pm\left(1+\sqrt{2}\right))</math> or <math>(\pm\left(1+\sqrt{2}\right), \pm 1)</math>. | ||
+ | |||
+ | By symmetry, <math>B_{1}B_{3}B_{5}B_{7}</math> is a square. Thus lines <math>\overleftrightarrow{B_{1}B_{3}}</math> and <math>\overleftrightarrow{B_{5}B_{7}}</math> are parallel, and its side length is the distance between these two lines. However, this is given to be the side length of the octagon, or <math>2</math>. | ||
+ | |||
+ | Suppose the common slope of the lines is <math>m</math> and let <math>m=\tan\theta</math>. Then, we want to find <cmath>\cos 2\left(90-\theta\right)=2\cos^{2}\left(90-\theta\right)-1=2\sin^{2}\theta-1.</cmath> | ||
+ | |||
+ | It can easily be seen that the equations of the lines are <cmath> | ||
+ | |||
+ | |||
+ | ==Solution 2== | ||
+ | |||
+ | Let <math>\theta=\angle M_1 M_3 B_1</math>. Thus we have that <math>\cos 2 \angle A_3 M_3 B_1=\cos \left(2\theta + \frac{\pi}{2} \right)=-\sin2\theta</math>. | ||
+ | |||
+ | Since <math>A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8</math> is a regular octagon and <math>B_1 B_3 = A_1 A_2</math>, let <math>k=A_1 A_2 = A_2 A_3 = B_1 B_3</math>. | ||
+ | |||
+ | |||
+ | Extend <math>\overline{A_1 A_2}</math> and <math>\overline{A_3 A_4}</math> until they intersect. Denote their intersection as <math>I_1</math>. Through similar triangles & the <math>45-45-90</math> triangles formed, we find that <math>M_1 M_3=\frac{k}{2}(2+\sqrt2)</math>. | ||
+ | |||
+ | We also have that <math>\triangle M_7 B_7 M_1 =\triangle M_1 B_1 M_3</math> through ASA congruence (<math>\angle B_7 M_7 M_1 =\angle B_1 M_1 M_3</math>, <math>M_7 M_1 = M_1 M_3</math>, <math>\angle B_7 M_1 M_7 =\angle B_1 M_3 M_1</math>). Therefore, we may let <math>n=M_1 B_7 = M_3 B_1</math>. | ||
+ | |||
+ | Thus, we have that <math>\sin\theta=\frac{n-k}{\frac{k}{2}(2+\sqrt2)}</math> and that <math>\cos\theta=\frac{n}{\frac{k}{2}(2+\sqrt2)}</math>. Therefore <math>\cos\theta-\sin\theta=\frac{k}{\frac{k}{2}(2+\sqrt2)}=\frac{2}{2+\sqrt2}=2-\sqrt2</math>. | ||
+ | |||
+ | Squaring gives that <math>\sin^2\theta - 2\sin\theta\cos\theta + \cos^2\theta = 6-4\sqrt2</math> and consequently that <math>-2\sin\theta\cos\theta = 5-4\sqrt2 = -\sin2\theta</math> through the identities <math>\sin^2\theta + \cos^2\theta = 1</math> and <math>\sin2\theta = 2\sin\theta\cos\theta</math>. | ||
+ | |||
+ | Thus we have that <math>\cos 2 \angle A_3 M_3 B_1=5-4\sqrt2=5-\sqrt{32}</math>. Therefore <math>m+n=5+32=\boxed{037}</math>. | ||
+ | |||
+ | ==Solution 3== | ||
+ | Let <math>A_1A_2 = 2</math>. Then <math>B_1</math> and <math>B_3</math> are the projections of <math>M_1</math> and <math>M_5</math> onto the line <math>B_1B_3</math>, so <math>2=B_1B_3=-M_1M_5\cos x</math>, where <math>x = \angle A_3M_3B_1</math>. Then since <math>M_1M_5 = 2+2\sqrt{2}, \cos x = \dfrac{-2}{2+2\sqrt{2}}= 1-\sqrt{2}</math>, <math>\cos 2x = 2\cos^2 x -1 = 5 - 4\sqrt{2} = 5-\sqrt{32}</math>, and <math>m+n=\boxed{037}</math>. | ||
+ | |||
+ | ==Solution 4== | ||
+ | Notice that <math>R_3</math> and <math>R_7</math> are parallel (<math>B_1B_3B_5B_7</math> is a square by symmetry and since the rays are perpendicular) and <math>B_1B_3=B_3B_5=s=</math> the distance between the parallel rays. If the regular hexagon as a side length of <math>s</math>, then <math>M_3M_7</math> has a length of <math>s+s\sqrt{2}</math>. Let <math>X</math> be on <math>R_3</math> such that <math>M_7X</math> is perpendicular to <math>M_3X</math>, and <math>\phi=\angle M_7M_3X</math>. The distance between <math>R_3</math> and <math>R_7</math> is <math>s=M_7X</math>, so <math>\sin\phi=\frac{s}{s+s\sqrt{2}}=\frac{1}{1+\sqrt{2}}</math>. | ||
+ | |||
+ | Since we are considering a regular hexagon, <math>M_3</math> is directly opposite to <math>M_7</math> and <math>\angle A_3M_3B_1=90 ^\circ +\phi</math>. All that's left is to calculate <math>\cos 2\angle A_3M_3B_1=\cos^2(90^\circ+\phi)-\sin^2(90^\circ+\phi)=\sin^2\phi-\cos^2\phi</math>. By drawing a right triangle or using the Pythagorean identity, <math>\cos^2\phi=\frac{2+2\sqrt2}{3+2\sqrt2}</math> and <math>\cos 2\angle A_3M_3B_1=\frac{-1-2\sqrt2}{3+2\sqrt2}=5-4\sqrt2=5-\sqrt{32}</math>, so <math>m+n=\boxed{037}</math>. | ||
+ | |||
+ | |||
+ | ==Solution 5== | ||
+ | Assume that <math>A_1A_2=1.</math> | ||
+ | Denote the center <math>O</math>, and the midpoint of <math>B_1</math> and <math>B_3</math> as <math>B_2</math>. Then we have that<cmath>\cos\angle A_3M_3B_1=\cos(\angle A_3M_3O+\angle OM_3B_1)=-\sin(\angle OM_3B_1)=-\frac{OB_2}{OM_3}=-\frac{1/2}{1/2+\sqrt2/2}=-\frac{1}{\sqrt2+1}=1-\sqrt2.</cmath>Thus, by the cosine double-angle theorem,<cmath>\cos2\angle A_3M_3B_1=2(1-\sqrt2)^2-1=5-\sqrt{32},</cmath>so <math>m+n=\boxed{037}</math>. | ||
+ | |||
+ | |||
+ | ==Video Solution== | ||
+ | [https://youtu.be/M47eLr9756A?si=b7aiOdKt5ZgnZY9h 2011 AIME I #14] | ||
+ | |||
+ | [https://mathproblemsolvingskills.wordpress.com/ MathProblemSolvingSkills.com] | ||
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== See also == | == See also == | ||
{{AIME box|year=2011|n=I|num-b=13|num-a=15}} | {{AIME box|year=2011|n=I|num-b=13|num-a=15}} | ||
+ | |||
+ | [[Category:Intermediate Geometry Problems]] | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 15:07, 2 September 2024
Contents
[hide]Problem
Let be a regular octagon. Let , , , and be the midpoints of sides , , , and , respectively. For , ray is constructed from towards the interior of the octagon such that , , , and . Pairs of rays and , and , and , and and meet at , , , respectively. If , then can be written in the form , where and are positive integers. Find .
Diagram
All distances are to scale.
Solution 1
We use coordinates. Let the octagon have side length and center . Then all of its vertices have the form or .
By symmetry, is a square. Thus lines and are parallel, and its side length is the distance between these two lines. However, this is given to be the side length of the octagon, or .
Suppose the common slope of the lines is and let . Then, we want to find
It can easily be seen that the equations of the lines are By the distance between parallel lines formula, a corollary of the point to line distance formula, the distance between these two lines is Since we want this to equal , we have Since we have . Thus The answer is .
Solution 2
Let . Thus we have that .
Since is a regular octagon and , let .
Extend and until they intersect. Denote their intersection as . Through similar triangles & the triangles formed, we find that .
We also have that through ASA congruence (, , ). Therefore, we may let .
Thus, we have that and that . Therefore .
Squaring gives that and consequently that through the identities and .
Thus we have that . Therefore .
Solution 3
Let . Then and are the projections of and onto the line , so , where . Then since , , and .
Solution 4
Notice that and are parallel ( is a square by symmetry and since the rays are perpendicular) and the distance between the parallel rays. If the regular hexagon as a side length of , then has a length of . Let be on such that is perpendicular to , and . The distance between and is , so .
Since we are considering a regular hexagon, is directly opposite to and . All that's left is to calculate . By drawing a right triangle or using the Pythagorean identity, and , so .
Solution 5
Assume that Denote the center , and the midpoint of and as . Then we have thatThus, by the cosine double-angle theorem,so .
Video Solution
See also
2011 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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