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Difference between revisions of "2011 AIME I Problems/Problem 14"
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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>.
 
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>.
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Alternate Solution:
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Let A_1A_2 = 2.  Then B_1 and B_3 are the projections of M_1 and M_5 onto the line B_1B_3, so 2=B_1B_3=(M_1M_5)(cos x), where x = angle A_3M_3B_1.  Then since M_1M_5 = 2+ sqrt 2, cos x = 2/(2+sqrt 2)= sqrt 2 -1, cos 2x = 2*(cos x)^2 -1 = 5 - 4 sqrt 2 = 5- sqrt 32, m+n=37.
  
 
== 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}}

Revision as of 13:22, 18 May 2011

Problem

Let $A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8$ be a regular octagon. Let $M_1$, $M_3$, $M_5$, and $M_7$ be the midpoints of sides $\overline{A_1 A_2}$, $\overline{A_3 A_4}$, $\overline{A_5 A_6}$, and $\overline{A_7 A_8}$, respectively. For $i = 1, 3, 5, 7$, ray $R_i$ is constructed from $M_i$ towards the interior of the octagon such that $R_1 \perp R_3$, $R_3 \perp R_5$, $R_5 \perp R_7$, and $R_7 \perp R_1$. Pairs of rays $R_1$ and $R_3$, $R_3$ and $R_5$, $R_5$ and $R_7$, and $R_7$ and $R_1$ meet at $B_1$, $B_3$, $B_5$, $B_7$ respectively. If $B_1 B_3 = A_1 A_2$, then $\cos 2 \angle A_3 M_3 B_1$ can be written in the form $m - \sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$.

Solution

Let $\theta=\angle M_1 M_3 B_1$. Thus we have that $\cos 2 \angle A_3 M_3 B_1=\cos(2\theta + \frac{\pi}{2})=-\sin2\theta$.

Since $A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8$ is a regular octagon and $B_1 B_3 = A_1 A_2$, let $k=A_1 A_2 = A_2 A_3 = B_1 B_3$.


Extend $\overline{A_1 A_2}$ and $\overline{A_5 A_6}$ until they intersect. Denote their intersection as $I_1$. Through similar triangles & the $45-45-90$ triangles formed, we find that $M_1 M_3=\frac{k}{2}(2+\sqrt2)$.

We also have that$\triangle M_7 B_7 M_1 =\triangle M_1 B_1 M_3$ through ASA congruence ($\angle B_7 M_7 M_1 =\angle B_1 M_1 M_3$, $M_7 M_1 = M_1 M_3$, $\angle B_7 M_1 M_7 =\angle B_1 M_3 M_1$). Therefore, we may let $n=M_1 B_7 = M_3 B_1$.

Thus, we have that $\sin\theta=\frac{n+k}{\frac{k}{2}(2+\sqrt2)}$ and that $\cos\theta=\frac{n}{\frac{k}{2}(2+\sqrt2)}$. Therefore $\sin\theta-\cos\theta=\frac{k}{\frac{k}{2}(2+\sqrt2)}=\frac{2}{2+\sqrt2}=2-\sqrt2$.

Squaring gives that $\sin^2\theta - 2\sin\theta\cos\theta + \cos^2\theta = 6-4\sqrt2$ and consequently that $-2\sin\theta\cos\theta = 5-4\sqrt2 = -\sin2\theta$ through the identities $\sin^2\theta + \cos^2\theta = 1$ and $\sin2\theta = 2\sin\theta\cos\theta$.

Thus we have that $\cos 2 \angle A_3 M_3 B_1=5-4\sqrt2=5-\sqrt{32}$. Therefore $m+n=5+32=\boxed{037}$.

Alternate Solution: Let A_1A_2 = 2. Then B_1 and B_3 are the projections of M_1 and M_5 onto the line B_1B_3, so 2=B_1B_3=(M_1M_5)(cos x), where x = angle A_3M_3B_1. Then since M_1M_5 = 2+ sqrt 2, cos x = 2/(2+sqrt 2)= sqrt 2 -1, cos 2x = 2*(cos x)^2 -1 = 5 - 4 sqrt 2 = 5- sqrt 32, m+n=37.

See also

2011 AIME I (Problems • Answer Key • Resources)
Preceded by
Problem 13 		Followed by
Problem 15
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All AIME Problems and Solutions
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