Difference between revisions of "2017 AMC 12A Problems/Problem 25"

Revision as of 15:10, 18 January 2020

Problem

The vertices $V$ of a centrally symmetric hexagon in the complex plane are given by $V=\left\{ \sqrt{2}i,-\sqrt{2}i, \frac{1}{\sqrt{8}}(1+i),\frac{1}{\sqrt{8}}(-1+i),\frac{1}{\sqrt{8}}(1-i),\frac{1}{\sqrt{8}}(-1-i) \right\}.$ For each $j$ , $1\leq j\leq 12$ , an element $z_j$ is chosen from $V$ at random, independently of the other choices. Let $P={\prod}_{j=1}^{12}z_j$ be the product of the $12$ numbers selected. What is the probability that $P=-1$ ?

$\textbf{(A) } \dfrac{5\cdot11}{3^{10}} \qquad \textbf{(B) } \dfrac{5^2\cdot11}{2\cdot3^{10}} \qquad \textbf{(C) } \dfrac{5\cdot11}{3^{9}} \qquad \textbf{(D) } \dfrac{5\cdot7\cdot11}{2\cdot3^{10}} \qquad \textbf{(E) } \dfrac{2^2\cdot5\cdot11}{3^{10}}$

Solution 1

It is possible to solve this problem using elementary counting methods. This solution proceeds by a cleaner generating function.

We note that $\pm \sqrt{2}i$ both lie on the imaginary axis and each of the $\frac{1}{\sqrt{8}}(\pm 1\pm i)$ have length $\frac{1}{2}$ and angle of odd multiples of $\pi/4$ , i.e. $\pi/4,3\pi/4,5\pi/4,7\pi/4$ . When we draw these 6 complex numbers out on the complex plane, we get a crystal-looking thing. Note that the total number of ways to choose 12 complex numbers is $6^{12}$ . Now we count the number of good combinations.

We first consider the lengths. When we multiply 12 complex numbers together, their magnitudes multiply. Suppose we have $n$ of the numbers $\pm \sqrt{2}i$ ; then we must have $\left(\sqrt{2}\right)^n\cdot\left(\frac{1}{2}\right)^{12-n}=1 \Longrightarrow n=8$ . Having $n=8$ will take care of the length of the product; now we need to deal with the angle.

We require $\sum\theta\equiv\pi \mod 2\pi$ . Letting $z$ be $e^{i\pi/4}$ , we see that the angles we have available are $\{z^1,z^2,z^3,z^5,z^6,z^7\}$ , where we must choose exactly 8 angles from the set $\{z^2,z^6\}$ and exactly 4 from the set $\{z^1,z^3,z^5,z^7\}$ . If we found a good combination where we had $a_i$ of each angle $z^i$ , then the amount this would contribute to our count would be $\binom{12}{4,8}\cdot\binom{8}{a_2,a_6}\cdot\binom{4}{a_1,a_3,a_5,a_7}$ . We want to add these all up. We proceed by generating functions.

Consider $(t_2x^2+t_6x^6)^8(t_1x^1+t_3x^3+t_5x^5+t_7x^7)^4.$ The expansion will be of the form $\sum_i\left(\sum_{\sum a=i} \binom{8}{a_2,a_6}\binom{4}{a_1,a_3,a_5,a_7}{t_1}^{a_1}{t_2}^{a_2}{t_3}^{a_3}{t_5}^{a_5}{t_6}^{a_6}{t_7}^{a_7}x^i \right)$ . Note that if we reduced the powers of $x$ mod $8$ and fished out the coefficient of $x^4$ and plugged in $t_i=1\ \forall\,i$ (and then multiplied by $\binom{12}{4,8}$ ) then we would be done. Since plugging in $t_i=1$ doesn't affect the $x$ 's, we do that right away. The expression then becomes $x^{20}(1+x^4)^8(1+x^2+x^4+x^6)^4=x^{20}(1+x^4)^{12}(1+x^2)^4=x^4(1+x^4)^{12}(1+x^2)^4,$ where the last equality is true because we are taking the powers of $x$ mod $8$ . Let $[x^n]f(x)$ denote the coefficient of $x^n$ in $f(x)$ . Note $[x^4] x^4(1+x^4)^{12}(1+x^2)^4=[x^0](1+x^4)^{12}(1+x^2)^4$ . We use the roots of unity filter, which states $\text{terms of }f(x)\text{ that have exponent congruent to }k\text{ mod }n=\frac{1}{n}\sum_{m=1}^n \frac{f(z^mx)}{z^{mk}},$ where $z=e^{i\pi/n}$ . In our case $k=0$ , so we only need to find the average of the $f(z^mx)$ 's. $\begin{align*} z^0 &\Longrightarrow (1+x^4)^{12}(1+x^2)^4,\\ z^1 &\Longrightarrow (1-x^4)^{12}(1+ix^2)^4,\\ z^2 &\Longrightarrow (1+x^4)^{12}(1-x^2)^4,\\ z^3 &\Longrightarrow (1-x^4)^{12}(1-ix^2)^4,\\ z^4 &\Longrightarrow (1+x^4)^{12}(1+x^2)^4,\\ z^5 &\Longrightarrow (1-x^4)^{12}(1+ix^2)^4,\\ z^6 &\Longrightarrow (1+x^4)^{12}(1-x^2)^4,\\ z^7 &\Longrightarrow (1-x^4)^{12}(1-ix^2)^4. \end{align*}$ We plug in $x=1$ and take the average to find the sum of all coefficients of $x^{\text{multiple of 8}}$ . Plugging in $x=1$ makes all of the above zero except for $z^0$ and $z^4$ . Averaging, we get $2^{14}$ . Now the answer is simply $\frac{\binom{12}{4,8}}{6^{12}}\cdot 2^{14}=\boxed{\frac{2^2\cdot 5\cdot 11}{3^{10}}}.$

Solution 2

By changing $z_1$ to $-z_1$ , we can give a bijection between cases where $P=-1$ and cases where $P=1$ , so we'll just find the probability that $P=\pm 1$ and divide by $2$ in the end. Multiplying the hexagon's vertices by $i$ doesn't change $P$ , and switching any $z_j$ with $-z_j$ doesn't change the property $P=\pm 1$ , so the probability that $P=\pm1$ remains the same if we only select our $z_j$ 's at random from $\left\{a= \sqrt 2,\quad b=\frac1{\sqrt{8}}(1+i),\quad c=\frac1{\sqrt{8}}(1-i)\right\}.$ Since $|a|=\sqrt2$ and $|b|=|c|=\frac12$ , we must choose $a$ exactly $8$ times to make $|P|=1$ . To ensure $P$ is real, we must either choose $b$ $4$ times, $c$ $4$ times, or both $b$ and $c$ $2$ times. This gives us a total of $2\binom{12}{4}+\binom{12}{2}\binom{10}{2}=(12\cdot 11\cdot 10\cdot 9)\left(\frac1{12}+\frac14\right)=(2^3\cdot 3^3\cdot 5\cdot 11)\frac13$ good sequences $z_1,\dots,z_{12}$ , and hence the final result is $\frac12\cdot \frac{2^3\cdot 3^2\cdot 5\cdot 11}{3^{12}}=\boxed{(E)}.$

Solution 3

We use generating functions and a roots of unity filter. Notice that all values in $V$ are eighth roots of unity multiplied by a constant. Let $x$ be a primitive eighth root of unity ( $e^{\frac{i\pi}{4}}$ ). The numbers in $V$ are then $\left \{ \frac{x}{2},x^2\sqrt2,\frac{x^3}{2},\frac{x^5}{2},x^6\sqrt2,\frac{x^7}{2} \right\}$ . To have $P=-1$ , we must have that $|P|=1$ , so eight of the $(z_i)$ must belong to $\left \{ x^2\sqrt2,x^6\sqrt2 \right\}$ and the other four must belong to $\left \{ \frac{x}{2},\frac{x^3}{2},\frac{x^5}{2},\frac{x^7}{2} \right\}$ So, we write the generating function $\left (x^2 +x^6 \right )^8 \left (x+x^3+x^5+x^7 \right )^4$ to describe the product. Note that this assumes that the $(z_i)$ that belong to $\left \{ x^2\sqrt2,x^6\sqrt2 \right\}$ come first, so we will need to multiply by $\binom{12}{4}$ at the end. We now apply a roots of unity filter to find the sum of the coefficients of the exponents that are $4\pmod{8}$ , or equivalently the coefficients of the powers that are multiples of $8$ of the following function: $P(x)=\left (x^2 +x^6 \right )^8 \left (x+x^3+x^5+x^7 \right )^4x^4$ Let $\zeta=e^{\frac{i\pi}{4}}$ . We are looking for $\frac{P(1)+P(\zeta)+P(\zeta^2)+P(\zeta^3)+P(\zeta^4)+P(\zeta^5)+P(\zeta^6)+P(\zeta^7)}{8}$ . $P(1)=P(-1)=2^{16}$ , and all of the rest are equal to $0$ . So, we get an answer of $\frac{2^{16}+2^{16}}{8\cdot 6^{12}}=\frac{2^{14}}{6^{12}}=\frac{2^2}{3^{10}}$ . But wait! We need to multiply by ${12}\choose {4}$ = $\frac{12\cdot11\cdot10\cdot9}{4\cdot3\cdot3\cdot1}=5\cdot9\cdot11$ . So, the answer is $\boxed{\text{\textbf{(E) }} \frac{2^2\cdot5\cdot11}{3^{10}} }$

@@ Line 7: / Line 7: @@
 <math>\textbf{(A) } \dfrac{5\cdot11}{3^{10}} \qquad \textbf{(B) } \dfrac{5^2\cdot11}{2\cdot3^{10}} \qquad \textbf{(C) } \dfrac{5\cdot11}{3^{9}} \qquad \textbf{(D) } \dfrac{5\cdot7\cdot11}{2\cdot3^{10}} \qquad \textbf{(E) } \dfrac{2^2\cdot5\cdot11}{3^{10}}</math>
-==Solution==
+==Solution 1==
 It is possible to solve this problem using elementary counting methods. This solution proceeds by a cleaner generating function.
-We note that <math>\pm \sqrt{2}i</math> both lie on the imaginary axis and each of the <math>\frac{1}{\sqrt{8}}(\pm 1\pm i)</math> have length <math>\frac{1}{2}</math> and angle of odd multiples of <math>\pi/4</math>, i.e. <math>\pi/4,3\pi/4,5\pi,4,7\pi/4</math>. When we draw these 6 complex numbers out on the complex plane, we get a crystal-looking thing. Note that the total number of ways to choose 12 complex numbers is <math>6^{12}</math>. Now we count the number of good combinations.
+We note that <math>\pm \sqrt{2}i</math> both lie on the imaginary axis and each of the <math>\frac{1}{\sqrt{8}}(\pm 1\pm i)</math> have length <math>\frac{1}{2}</math> and angle of odd multiples of <math>\pi/4</math>, i.e. <math>\pi/4,3\pi/4,5\pi/4,7\pi/4</math>. When we draw these 6 complex numbers out on the complex plane, we get a crystal-looking thing. Note that the total number of ways to choose 12 complex numbers is <math>6^{12}</math>. Now we count the number of good combinations.
 We first consider the lengths. When we multiply 12 complex numbers together, their magnitudes multiply. Suppose we have <math>n</math> of the numbers <math>\pm \sqrt{2}i</math>; then we must have <math>\left(\sqrt{2}\right)^n\cdot\left(\frac{1}{2}\right)^{12-n}=1 \Longrightarrow n=8</math>. Having <math>n=8</math> will take care of the length of the product; now we need to deal with the angle.

2017 AMC 12A (Problems • Answer Key • Resources)
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