Difference between revisions of "2009 AIME II Problems/Problem 13"

Revision as of 21:46, 20 July 2020

Problem

Let $A$ and $B$ be the endpoints of a semicircular arc of radius $2$ . The arc is divided into seven congruent arcs by six equally spaced points $C_1$ , $C_2$ , $\dots$ , $C_6$ . All chords of the form $\overline {AC_i}$ or $\overline {BC_i}$ are drawn. Let $n$ be the product of the lengths of these twelve chords. Find the remainder when $n$ is divided by $1000$ .

Solution

Solution 1

Let the radius be 1 instead. All lengths will be halved so we will multiply by $2^{12}$ at the end. Place the semicircle on the complex plane, with the center of the circle being 0 and the diameter being the real axis. Then $C_1,\ldots, C_6$ are 6 of the 14th roots of unity. Let $\omega=\text{cis}\frac{360^{\circ}}{14}$ ; then $C_1,\ldots, C_6$ correspond to $\omega,\ldots, \omega^6$ . Let $C_1',\ldots, C_6'$ be their reflections across the diameter. These points correspond to $\omega^8\ldots, \omega^{13}$ . Then the lengths of the segments are $|1-\omega|,\ldots, |1-\omega^6|,|1-\omega^8|,\ldots |1-\omega^{13}|$ . Noting that $B$ represents 1 in the complex plane, the desired product is $\begin{align*} BC_1\cdots BC_6 \cdot AC_1\cdots AC_6&= BC_1\cdots BC_6 \cdot BC_1'\cdots BC_6\\ &= |(x-\omega^1)\ldots(x-\omega^6)(x-\omega^8)\ldots(x-\omega^{13})| \end{align*}$

for $x=1$ . However, the polynomial $(x-\omega^1)\ldots(x-\omega^6)(x-\omega^8)\ldots(x-\omega^{13})$ has as its zeros all 14th roots of unity except for $-1$ and $1$ . Hence $(x-\omega^1)\ldots(x-\omega^6)(x-\omega^8)\ldots(x-\omega^{13})=\frac{x^{14}-1}{(x-1)(x+1)}=x^{12}+x^{10}+\cdots +x^2+1.$ Thus the product is $|x^{12}+\cdots +x^2+1|=7$ ( $x=1$ ) when the radius is 1, and the product is $2^{12}\cdot 7=28672$ . Thus the answer is $\boxed {672}$ .

Solution 2

Let $O$ be the midpoint of $A$ and $B$ . Assume $C_1$ is closer to $A$ instead of $B$ . $\angle AOC_1$ = $\frac {\pi}{7}$ . Using the Law of Cosines,

$\overline {AC_1}^2$ = $8 - 8 \cos \frac {\pi}{7}$ , $\overline {AC_2}^2$ = $8 - 8 \cos \frac {2\pi}{7}$ , . . . $\overline {AC_6}^2$ = $8 - 8 \cos \frac {6\pi}{7}$

So $n$ = $(8^6)(1 - \cos \frac {\pi}{7})(1 - \cos \frac {2\pi}{7})\dots(1 - \cos \frac{6\pi}{7})$ . It can be rearranged to form

$n$ = $(8^6)(1 - \cos \frac {\pi}{7})(1 - \cos \frac {6\pi}{7})\dots(1 - \cos \frac {3\pi}{7})(1 - \cos \frac {4\pi}{7})$ .

Since $\cos a = - \cos (\pi - a)$ , so we have

$n$ = $(8^6)(1 - \cos \frac {\pi}{7})(1 + \cos \frac {\pi}{7}) \dots (1 - \cos \frac {3\pi}{7})(1 + \cos \frac {3\pi}{7})$

= $(8^6)(1 - \cos^2 \frac {\pi}{7})(1 - \cos^2 \frac {2\pi}{7})(1 - \cos^2 \frac {3\pi}{7})$

= $(8^6)(\sin^2 \frac {\pi}{7})(\sin^2 \frac {2\pi}{7})(\sin^2 \frac {3\pi}{7})$

It can be shown that $\sin \frac {\pi}{7} \sin \frac {2\pi}{7} \sin \frac {3\pi}{7}$ = $\frac {\sqrt {7}}{8}$ , so $n$ = $8^6(\frac {\sqrt {7}}{8})^2$ = $7(8^4)$ = $28672$ , so the answer is $\boxed {672}$

Solution 3

Note that for each $k$ the triangle $ABC_k$ is a right triangle. Hence the product $AC_k \cdot BC_k$ is twice the area of the triangle $ABC_k$ . Knowing that $AB=4$ , the area of $ABC_k$ can also be expressed as $2c_k$ , where $c_k$ is the length of the altitude from $C_k$ onto $AB$ . Hence we have $AC_k \cdot BC_k = 4c_k$ .

By the definition of $C_k$ we obviously have $c_k = 2\sin\frac{k\pi}7$ .

From these two observations we get that the product we should compute is equal to $8^6 \cdot \prod_{k=1}^6 \sin \frac{k\pi}7$ , which is the same identity as in Solution 2.

Computing the product of sines

In this section we show one way how to evaluate the product $\prod_{k=1}^6 \sin \frac{k\pi}7 = \prod_{k=1}^3 (\sin \frac{k\pi}7)^2$ .

Let $\omega_k = \cos \frac{2k\pi}7 + i\sin \frac{2k\pi}7$ . The numbers $1,\omega_1,\omega_2,\dots,\omega_6$ are the $7$ -th complex roots of unity. In other words, these are the roots of the polynomial $x^7-1$ . Then the numbers $\omega_1,\omega_2,\dots,\omega_6$ are the roots of the polynomial $\frac{x^7-1}{x-1} = x^6+x^5+\cdots+x+1$ .

We just proved the identity $\prod_{k=1}^6 (x - \omega_k) = x^6+x^5+\cdots+x+1$ . Substitute $x=1$ . The right hand side is obviously equal to $7$ . Let's now examine the left hand side. We have:

$\begin{align*} (1-\omega_k)(1-\omega_{k-7})=|1-\omega_k|^2 & = \left( 1-\cos \frac{2k\pi}7 \right)^2 + \left( \sin \frac{2k\pi}7 \right)^2 \\ & = 2-2\cos \frac{2k\pi}7 \\ & = 2-2 \left( 1 - 2 \left( \sin \frac{k\pi}7 \right)^2 \right) \\ & = 4\left( \sin \frac{k\pi}7 \right)^2 \end{align*}$

Therefore the size of the left hand side in our equation is $\prod_{k=1}^3 4 (\sin \frac{k\pi}7)^2 = 2^6 \prod_{k=1}^3 (\sin \frac{k\pi}7)^2$ . As the right hand side is $7$ , we get that $\prod_{k=1}^3 (\sin \frac{k\pi}7)^2 = \frac{7}{2^6}$ .

@@ Line 40: / Line 40: @@
 <math>n</math> = <math>(8^6)(1 - \cos \frac {\pi}{7})(1 - \cos \frac {6\pi}{7})\dots(1 - \cos \frac {3\pi}{7})(1 - \cos \frac {4\pi}{7})</math>.
-<math>\cos a</math> = - <math>\cos (\pi - a)</math>, so we have
+Since <math>\cos a = - \cos (\pi - a)</math>, so we have
 <math>n</math> = <math>(8^6)(1 - \cos \frac {\pi}{7})(1 + \cos \frac {\pi}{7}) \dots (1 - \cos \frac {3\pi}{7})(1 + \cos \frac {3\pi}{7})</math>

2009 AIME II (Problems • Answer Key • Resources)
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