Difference between revisions of "2012 AIME II Problems/Problem 13"

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Problem 13

Equilateral $\triangle ABC$ has side length $\sqrt{111}$ . There are four distinct triangles $AD_1E_1$ , $AD_1E_2$ , $AD_2E_3$ , and $AD_2E_4$ , each congruent to $\triangle ABC$ , with $BD_1 = BD_2 = \sqrt{11}$ . Find $\sum_{k=1}^4(CE_k)^2$ .

Solution 1

Note that there are only two possible locations for points $D_1$ and $D_2$ , as they are both $\sqrt{111}$ from point $A$ and $\sqrt{11}$ from point $B$ , so they are the two points where a circle centered at $A$ with radius $\sqrt{111}$ and a circle centered at $B$ with radius $\sqrt{11}$ intersect. Let $D_1$ be the point on the opposite side of $\overline{AB}$ from $C$ , and $D_2$ the point on the same side of $\overline{AB}$ as $C$ .

Let $\theta$ be the measure of angle $BAD_1$ (which is also the measure of angle $BAD_2$ ); by the Law of Cosines, $\begin{align*}\sqrt{11}^2 &= \sqrt{111}^2 + \sqrt{111}^2 - 2 \cdot \sqrt{111} \cdot \sqrt{111} \cdot \cos\theta\\ 11 &= 222(1 - \cos\theta)\end{align*}$

There are two equilateral triangles with $\overline{AD_1}$ as a side; let $E_1$ be the third vertex that is farthest from $C$ , and $E_2$ be the third vertex that is nearest to $C$ .

Angle $E_1AC = E_1AD_1 + D_1AB + BAC = 60 + \theta + 60 = 120 + \theta$ ; by the Law of Cosines, $\begin{align*}(E_1C)^2 &= (E_1A)^2 + (AC)^2 - 2 (E_1A) (E_1C)\cos(120 + \theta)\\ &= 111 + 111 - 222\cos(120 + \theta)\end{align*}$ Angle $E_2AC = \theta$ ; by the Law of Cosines, $\begin{align*}(E_2C)^2 &= (E_2A)^2 + (AC)^2 - 2 (E_2A) (E_2C)\cos\theta\\ &= 111 + 111 - 222\,\cos\theta\end{align*}$

There are two equilateral triangles with $\overline{AD_2}$ as a side; let $E_3$ be the third vertex that is farthest from $C$ , and $E_4$ be the third vertex that is nearest to $C$ .

Angle $E_3AC = E_3AB + BAC = (60 - \theta) + 60 = 120 - \theta$ ; by the Law of Cosines, $\begin{align*}(E_3C)^2 &= (E_3A)^2 + (AC)^2 - 2 (E_3A) (E_3C)\cos(120 - \theta)\\ &= 111 + 111 - 222\cos(120 - \theta)\end{align*}$ Angle $E_4AC = \theta$ ; by the Law of Cosines, $\begin{align*}(E_4C)^2 &= (E_4A)^2 + (AC)^2 - 2 (E_4A) (E_4C)\cos\theta \\ &= 111 + 111 - 222\cos\theta\end{align*}$

The solution is: $\begin{align*} \sum_{k=1}^4(CE_k)^2 &= (E_1C)^2 + (E_3C)^2 + (E_2C)^2 + (E_4C)^2\\ &= 222(1 - \cos(120 + \theta)) + 222(1 - \cos(120 - \theta)) + 222(1 - \cos\theta) + 222(1 - \cos\theta)\\ &= 222((1 - (\cos120\cos\theta - \sin120\sin\theta)) + (1 - (\cos120\cos\theta + \sin120\sin\theta)) + 2(1 -\cos\theta))\\ &= 222(1 - \cos120\cos\theta + \sin120\sin\theta + 1 - \cos120\cos\theta - \sin120\sin\theta + 2 - 2\cos\theta)\\ &= 222(1 + \frac{1}{2}\cos\theta + 1 + \frac{1}{2}\cos\theta + 2 - 2\cos\theta)\\ &= 222(4 - \cos\theta)\\ &= 666 + 222(1 - \cos\theta) \end{align*}$ Substituting $11$ for $222(1 - \cos\theta)$ gives the solution $666 + 11 = \framebox{677}.$

Solution 2

This problem is pretty much destroyed by complex plane geometry, which is similar to vector geometry only with the power of easy rotation. Place the triangle in the complex plane by letting $C$ be the origin, placing $B$ along the x-axis, and $A$ in the first quadrant. Let $r=\sqrt{111}$ . If $\omega$ denotes the sixth root of unity, $e^{i\pi/3}$ , then we have $C=0$ , $B=r$ , and $A=r\omega.$ Recall that counter-clockwise rotation in the complex plane by an angle $\theta$ is accomplished by multiplication by $e^{i\theta}$ (and clockwise rotation is multiplication by its conjugate). So, we can find $D_1$ and $D_2$ by rotating $B$ around $A$ by angles of $\theta$ and $-\theta$ , where $\theta$ is the apex angle in the isoceles triangle with sides $\sqrt{111}$ , $\sqrt{111}$ , and $\sqrt{11}$ . That is, let $z=e^{i\theta}$ , and then:

$D_1=A+z(B-A)$ , and $D_2=A+\overline{z}(B-A)$ . Now notice that $B-A=\overline{A}$ , so this simplifies further to:

$D_1=A+z\overline{A}$ , and $D_2=A+\overline{z}\overline{A}$ .

Similarly, we can write $E_1$ , $E_2$ , $E_3$ , and $E_4$ by rotating $D_1$ and $D_2$ around $A$ by $\pm\pi/3$ :

$E_1=A+\omega(D_1-A)$ , $E_2=A+\overline{\omega}(D_1-A)$ , $E_3=A+\omega(D_2-A)$ , $E_4=A+\overline{\omega}(D_2-A)$ . Thus:

$E_1=A+\omega z \overline{A}$ , $E_2=A+\overline{\omega} z \overline{A}$ , $E_3=A+\omega\overline{z}\overline{A}$ , $E_4=A+\overline{\omega}\overline{z}\overline{A}$ .

Now to find some magnitudes, which is easy since we chose $C$ as the origin:

$\|E_1\|^2=(A+\omega z \overline{A})(\overline{A}+\overline{\omega z}A)=2\|A\|^2+\omega z \overline{A}^2 + \overline{\omega}\overline{z}A^2$ ,

$\|E_2\|^2=(A+\overline{\omega} z \overline{A})(\overline{A}+\omega \overline{z}A)=2\|A\|^2+\overline{\omega} z \overline{A}^2 + \omega\overline{z}A^2$ ,

$\|E_3\|^2=(A+\omega \overline{z} \overline{A})(\overline{A}+\overline{\omega} z A)=2\|A\|^2+\omega \overline{z} \overline{A}^2 + \overline{\omega}zA^2$ ,

$\|E_4\|^2=(A+\overline{\omega z} \overline{A})(\overline{A}+\omega zA)=2\|A\|^2+\overline{\omega z} \overline{A}^2 + \omega zA^2$ .

Adding these up, the sum equals $8\|A\|^2+(\overline{A}^2+A^2)(\omega z + \omega \overline{z} + \overline{\omega}z + \overline{\omega}\overline{z}) = 8\|A\|^2+(\overline{A}^2+A^2)(z + \overline{z})( \omega+ \overline{\omega})$ .

(Isn't that nice?) Notice that $\overline{A}^2+A^2 = r^2(\overline{\omega}^2+\omega^2) = -r^2$ , and $\omega+ \overline{\omega}=1$ , so that this sum simplifies further to $888-111(z + \overline{z})$ .

Finally, $z + \overline{z} = 2\cos{\theta}$ , which is found using the law of cosines on that isoceles triangle: $11=111+111-222\cos{\theta}$ , so $2\cos{\theta}=211/111$ .

Thus, the sum equals $888-211=\framebox{677}$ .

Solution 3

This method uses complex numbers with $A$ as the origin. Let $A=0$ , $B=\sqrt{111}$ , $C = \sqrt{111}\theta$ , where $\theta = e^{i \pi/3} = \frac{1}{2} + \frac{\sqrt{3}}{2}i$ .

Also, let $x$ be $D_1$ or $D_2$ . Then

$|x|=\sqrt{111}, |x-\sqrt{111}|=\sqrt{11}$

Therefore, $11 = |x-\sqrt{111}|^2 = |x|^2 + 111 -2\sqrt{111}Re(x) = 222 - 2\sqrt{111}Re(x)$ , so

$2\sqrt{111}Re(x) = 211.$

Since $E_1$ , $E_2$ are one of $D_1\theta$ or $D_1\theta^{-1}$ , without loss of generality, let $E_1=D_1\theta$ and $E_2 = D_1\theta^{-1}$ . Then

$|CE_1|^2 = |\sqrt{111}-D_1|^2 = |\sqrt{111}-x|^2=11$ $|CE_2|^2 = |\theta^2 \sqrt{111} - D_1|^2 = 222-2\sqrt{111} Re(D_1 \theta^2)$

One can similarly get $|CE_3|=11$ and $|CE_4|=222-2\sqrt{111} Re(D_2 \theta^2)$ , so the desired sum is equal to

$22+444-2\sqrt{111}(Re(D_1 \theta^2)+Re(D_1 \theta^2))$

Note that $Re(D_{1,2} \theta^2) = Re((Re(x) \pm Im(x)i)(-1/2 + \sqrt{3}/2i)) = - (Re(x)/2 \pm Im(x)\sqrt{3}/2)$ , so the sum of these two is just $-Re(x)$ . Therefore the desired sum is equal to

$22+444+2\sqrt{111}Re(x) = 22+444+211 = \framebox{677}.$

@@ Line 58: / Line 58: @@
 <math>\|E_1\|^2=(A+\omega z \overline{A})(\overline{A}+\overline{\omega z}A)=2\|A\|^2+\omega z \overline{A}^2 + \overline{\omega}\overline{z}A^2</math>,
-<math>\|E_2\|^2=(A+\overline{\omega} z \overline{A})(\overline{A}+\omega} \overline{z}A)=2\|A\|^2+\overline{\omega} z \overline{A}^2 + \omega\overline{z}A^2</math>,
+<math>\|E_2\|^2=(A+\overline{\omega} z \overline{A})(\overline{A}+\omega \overline{z}A)=2\|A\|^2+\overline{\omega} z \overline{A}^2 + \omega\overline{z}A^2</math>,
 <math>\|E_3\|^2=(A+\omega \overline{z} \overline{A})(\overline{A}+\overline{\omega} z A)=2\|A\|^2+\omega \overline{z} \overline{A}^2 + \overline{\omega}zA^2</math>,

2012 AIME II (Problems • Answer Key • Resources)
Preceded by Problem 12	Followed by Problem 14
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