Difference between revisions of "2013 AMC 12B Problems/Problem 25"

(Created page with "==Problem== Let <math>G</math> be the set of polynomials of the form <cmath> P(z)=z^n+c_{n-1}z^{n-1}+\cdots+c_2z^2+c_1z+50, </cmath> where <math> c_1,c_2,\cdots, c_{n-1} </math>...")
 
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where <math> c_1,c_2,\cdots, c_{n-1} </math> are integers and <math>P(z)</math> has distinct roots of the form <math>a+ib</math> with <math>a</math> and <math>b</math> integers. How many polynomials are in <math>G</math>?
 
where <math> c_1,c_2,\cdots, c_{n-1} </math> are integers and <math>P(z)</math> has distinct roots of the form <math>a+ib</math> with <math>a</math> and <math>b</math> integers. How many polynomials are in <math>G</math>?
  
<math> \textbf{(A)}\ 288\qquad\textbf{(B)}\ 528\qquad\textbf{(C)}\ 576\qquad\textbf{(D}}\ 992\qquad\textbf{(E)}\ 1056 </math>
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<math> \textbf{(A)}\ 288\qquad\textbf{(B)}\ 528\qquad\textbf{(C)}\ 576\qquad\textbf{(D)}\ 992\qquad\textbf{(E)}\ 1056 </math>
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==Solution==
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===Solution 1===
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If we factor into irreducible polynomials (in <math>\mathbb{Q}[x]</math>), each factor <math>f_i</math> has exponent <math>1</math> in the factorization and degree at most <math>2</math> (since the <math>a+bi</math> with <math>b\ne0</math> come in conjugate pairs with product <math>a^2+b^2</math>). Clearly we want the product of constant terms of these polynomials to equal <math>50</math>; for <math>d\mid 50</math>, let <math>f(d)</math> be the number of permitted <math>f_i</math> with constant term <math>d</math>. It's easy to compute <math>f(1)=2</math>, <math>f(2)=3</math>, <math>f(5)=5</math>, <math>f(10)=5</math>, <math>f(25)=6</math>, <math>f(50)=7</math>, and obviously <math>f(d) = 1</math> for negative <math>d\mid 50</math>.
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Note that by the distinctness condition, the only constant terms <math>d</math> that can be repeated are those with <math>d^2\mid 50</math> and <math>f(d)>1</math>, i.e. <math>+1</math> and <math>+5</math>. Also, the <math>+1</math>s don't affect the product, so we can simply count the number of polynomials with no constant terms of <math>+1</math> and multiply by <math>2^{f(1)} = 4</math> at the end.
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We do casework on the (unique) even constant term <math>d\in\{\pm2,\pm10,\pm50\}</math> in our product. For convenience, let <math>F(d)</math> be the number of ways to get a product of <math>50/d</math> without using <math>\pm 1</math> (so only using <math>\pm5,\pm25</math>) and recall <math>f(-1) = 1</math>; then our final answer will be <math>2^{f(1)}\sum_{d\in\{2,10,50\}}(f(-d)+f(d))(F(-d)+F(d))</math>. It's easy to compute <math>F(-50)=0</math>, <math>F(50)=1</math>, <math>F(-10)=f(5)=5</math>, <math>F(10)=f(-5)=1</math>, <math>F(-2)=f(-25)+f(-5)f(5)=6</math>, <math>F(2)=f(25)+\binom{f(5)}{2}=16</math>, so we get
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<cmath> 4 [ (1+3)(6+16) + (1+5)(1+5) + (1+7)(0+1) ] = 4[132] = \boxed{\textbf{(B) }528} </cmath>
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===Solution 2===
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Disregard sign; we can tack on <math>x-1</math> if the product ends up being negative.
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<math>1: \pm i,-1</math> (2) (1 is not included)
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<math>2: \pm 2, \pm 1\pm i</math> (4)
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<math>5: \pm 2\pm i, \pm 1\pm 2i, \pm 5</math> (6)
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<math>10: \pm 3\pm i, \pm 1\pm 3i, \pm 10</math> (6)
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<math>25: \pm 25, \pm 3\pm 4i, \pm 4\pm 3i, \pm 5i</math> (7)
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<math>50: \pm 50, \pm 1\pm 7i, \pm7\pm i, \pm 5\pm 5i</math> (8)
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Our answer is <math>2^2\left(4\cdot\binom{6}{2}+6\cdot 6+4\cdot 7+8\right)=\boxed{528.}</math>
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== See also ==
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{{AMC12 box|year=2013|ab=B|num-b=24|after=Last Question}}
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{{MAA Notice}}

Latest revision as of 20:20, 28 December 2017

Problem

Let $G$ be the set of polynomials of the form \[P(z)=z^n+c_{n-1}z^{n-1}+\cdots+c_2z^2+c_1z+50,\] where $c_1,c_2,\cdots, c_{n-1}$ are integers and $P(z)$ has distinct roots of the form $a+ib$ with $a$ and $b$ integers. How many polynomials are in $G$?

$\textbf{(A)}\ 288\qquad\textbf{(B)}\ 528\qquad\textbf{(C)}\ 576\qquad\textbf{(D)}\ 992\qquad\textbf{(E)}\ 1056$

Solution

Solution 1

If we factor into irreducible polynomials (in $\mathbb{Q}[x]$), each factor $f_i$ has exponent $1$ in the factorization and degree at most $2$ (since the $a+bi$ with $b\ne0$ come in conjugate pairs with product $a^2+b^2$). Clearly we want the product of constant terms of these polynomials to equal $50$; for $d\mid 50$, let $f(d)$ be the number of permitted $f_i$ with constant term $d$. It's easy to compute $f(1)=2$, $f(2)=3$, $f(5)=5$, $f(10)=5$, $f(25)=6$, $f(50)=7$, and obviously $f(d) = 1$ for negative $d\mid 50$.

Note that by the distinctness condition, the only constant terms $d$ that can be repeated are those with $d^2\mid 50$ and $f(d)>1$, i.e. $+1$ and $+5$. Also, the $+1$s don't affect the product, so we can simply count the number of polynomials with no constant terms of $+1$ and multiply by $2^{f(1)} = 4$ at the end.

We do casework on the (unique) even constant term $d\in\{\pm2,\pm10,\pm50\}$ in our product. For convenience, let $F(d)$ be the number of ways to get a product of $50/d$ without using $\pm 1$ (so only using $\pm5,\pm25$) and recall $f(-1) = 1$; then our final answer will be $2^{f(1)}\sum_{d\in\{2,10,50\}}(f(-d)+f(d))(F(-d)+F(d))$. It's easy to compute $F(-50)=0$, $F(50)=1$, $F(-10)=f(5)=5$, $F(10)=f(-5)=1$, $F(-2)=f(-25)+f(-5)f(5)=6$, $F(2)=f(25)+\binom{f(5)}{2}=16$, so we get \[4 [ (1+3)(6+16) + (1+5)(1+5) + (1+7)(0+1) ] = 4[132] = \boxed{\textbf{(B) }528}\]

Solution 2

Disregard sign; we can tack on $x-1$ if the product ends up being negative.

$1: \pm i,-1$ (2) (1 is not included)

$2: \pm 2, \pm 1\pm i$ (4)

$5: \pm 2\pm i, \pm 1\pm 2i, \pm 5$ (6)

$10: \pm 3\pm i, \pm 1\pm 3i, \pm 10$ (6)

$25: \pm 25, \pm 3\pm 4i, \pm 4\pm 3i, \pm 5i$ (7)

$50: \pm 50, \pm 1\pm 7i, \pm7\pm i, \pm 5\pm 5i$ (8)

Our answer is $2^2\left(4\cdot\binom{6}{2}+6\cdot 6+4\cdot 7+8\right)=\boxed{528.}$

See also

2013 AMC 12B (ProblemsAnswer KeyResources)
Preceded by
Problem 24
Followed by
Last Question
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All AMC 12 Problems and Solutions

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