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Difference between revisions of "2004 AMC 12A Problems/Problem 23"

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<cmath>\sum_{k = 1}^{2004}{a_k} = \sum_{k = 1}^{2004}{b_k}.</cmath>
 
<cmath>\sum_{k = 1}^{2004}{a_k} = \sum_{k = 1}^{2004}{b_k}.</cmath>
  
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Which of the following quantities can be a nonzero number?
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Which of the following quantities can be a non zero number?
  
 
<math>\text {(A)} c_0 \qquad \text {(B)} c_{2003} \qquad \text {(C)} b_2b_3...b_{2004} \qquad \text {(D)} \sum_{k = 1}^{2004}{a_k} \qquad \text {(E)}\sum_{k = 1}^{2004}{c_k}</math>
 
<math>\text {(A)} c_0 \qquad \text {(B)} c_{2003} \qquad \text {(C)} b_2b_3...b_{2004} \qquad \text {(D)} \sum_{k = 1}^{2004}{a_k} \qquad \text {(E)}\sum_{k = 1}^{2004}{c_k}</math>
  
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== Solution ==
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== Solution 1==
 
We have to evaluate the answer choices and use process of elimination:
 
We have to evaluate the answer choices and use process of elimination:
  
 
*<math>\mathrm{(A)}</math>: We are given that <math>a_1 = b_1 = 0</math>, so <math>z_1 = 0</math>. If one of the roots is zero, then <math>P(0) = c_0 = 0</math>.  
 
*<math>\mathrm{(A)}</math>: We are given that <math>a_1 = b_1 = 0</math>, so <math>z_1 = 0</math>. If one of the roots is zero, then <math>P(0) = c_0 = 0</math>.  
 
*<math>\mathrm{(B)}</math>: By [[Vieta's formulas]], we know that <math>-\frac{c_{2003}}{c_{2004}}</math> is the sum of all of the roots of <math>P(x)</math>. Since that is real, <math> \sum_{k = 1}^{2004}{b_k}=0=\sum_{k = 1}^{2004}{a_k}</math>, and <math>\frac{c_{2003}}{c_{2004}}=0</math>, so <math>c_{2003}=0</math>.
 
*<math>\mathrm{(B)}</math>: By [[Vieta's formulas]], we know that <math>-\frac{c_{2003}}{c_{2004}}</math> is the sum of all of the roots of <math>P(x)</math>. Since that is real, <math> \sum_{k = 1}^{2004}{b_k}=0=\sum_{k = 1}^{2004}{a_k}</math>, and <math>\frac{c_{2003}}{c_{2004}}=0</math>, so <math>c_{2003}=0</math>.
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*<math>\mathrm{(C)}</math>: All of the coefficients are real. For sake of contradiction suppose none of <math>b_{2\ldots 2004}</math> are zero. Then for each complex root <math>z_i</math>, its [[complex conjugate]] <math>\overline{z_i} = a_i - b_ik</math> is also a root. So the roots should pair up, but we have an odd number of imaginary roots! (Remember that <math>b_1 = 0</math>.) This gives us the contradiction, and therefore the product is equal to zero.
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*<math>\mathrm{(C)}</math>: All of the coefficients are real. For sake of contradiction suppose none of <math>b_{2\ldots 2004}</math> are zero. Then for each complex root <math>z_k</math>, its [[complex conjugate]] <math>\overline{z_k} = a_k - b_k i</math> is also a root. So the roots should pair up, but we have an odd number of imaginary roots! (Remember that <math>b_1 = 0</math>.) This gives us the contradiction, and therefore the product is equal to zero.
 
*<math>\mathrm{(D)}</math>: We are given that <math>\sum_{k = 1}^{2004}{a_k} = \sum_{k = 1}^{2004}{b_k}</math>. Since the coefficients are real, it follows that if a root is complex, its conjugate is also a root; and the sum of the imaginary parts of complex conjugates is zero. Hence the RHS is zero.
 
*<math>\mathrm{(D)}</math>: We are given that <math>\sum_{k = 1}^{2004}{a_k} = \sum_{k = 1}^{2004}{b_k}</math>. Since the coefficients are real, it follows that if a root is complex, its conjugate is also a root; and the sum of the imaginary parts of complex conjugates is zero. Hence the RHS is zero.
  
 
There is, however, no reason to believe that <math>\boxed{\mathrm{E}}</math> should be zero (in fact, that quantity is <math>P(1)</math>, and there is no evidence that <math>1</math> is a root of <math>P(x)</math>).
 
There is, however, no reason to believe that <math>\boxed{\mathrm{E}}</math> should be zero (in fact, that quantity is <math>P(1)</math>, and there is no evidence that <math>1</math> is a root of <math>P(x)</math>).
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==Solution 2(cheap method using answer choices)==
 +
Rule out answer choices <math>A, B, D</math> as done in solution 1. Assume for the sake of contradiction that <math>\mathrm{(E)}</math> is <math>0</math>. Then <math>P(1)=0</math>, so there is some <math>m\neq 1</math> such that <math>z_m=1+0i</math>, which implies at least one <math>b_i</math>, <math>i\neq 1</math> is <math>0</math> SO if <math>\sum_{k=1}^{2004}c_k=0</math>, then <math>\prod_{i=2}^{2004}b_i=0</math>. So we have that <math>\sum_{k=1}^{2004}c_k\neq 0</math>, because then that would rule out all the answer choices. Hence <math>\boxed{\mathrm{E}}</math> must be nonzero, so the answer is <math>\mathrm{(E)}</math>.
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-vsamc
  
 
== See also ==
 
== See also ==

Latest revision as of 13:13, 5 September 2021

Problem

A polynomial

\[P(x) = c_{2004}x^{2004} + c_{2003}x^{2003} + ... + c_1x + c_0\]

has real coefficients with $c_{2004}\not = 0$ and $2004$ distinct complex zeroes $z_k = a_k + b_ki$, $1\leq k\leq 2004$ with $a_k$ and $b_k$ real, $a_1 = b_1 = 0$, and

\[\sum_{k = 1}^{2004}{a_k} = \sum_{k = 1}^{2004}{b_k}.\]

Which of the following quantities can be a non zero number?

$\text {(A)} c_0 \qquad \text {(B)} c_{2003} \qquad \text {(C)} b_2b_3...b_{2004} \qquad \text {(D)} \sum_{k = 1}^{2004}{a_k} \qquad \text {(E)}\sum_{k = 1}^{2004}{c_k}$

Solution 1

We have to evaluate the answer choices and use process of elimination:

  • $\mathrm{(A)}$: We are given that $a_1 = b_1 = 0$, so $z_1 = 0$. If one of the roots is zero, then $P(0) = c_0 = 0$.
  • $\mathrm{(B)}$: By Vieta's formulas, we know that $-\frac{c_{2003}}{c_{2004}}$ is the sum of all of the roots of $P(x)$. Since that is real, $\sum_{k = 1}^{2004}{b_k}=0=\sum_{k = 1}^{2004}{a_k}$, and $\frac{c_{2003}}{c_{2004}}=0$, so $c_{2003}=0$.
  • $\mathrm{(C)}$: All of the coefficients are real. For sake of contradiction suppose none of $b_{2\ldots 2004}$ are zero. Then for each complex root $z_k$, its complex conjugate $\overline{z_k} = a_k - b_k i$ is also a root. So the roots should pair up, but we have an odd number of imaginary roots! (Remember that $b_1 = 0$.) This gives us the contradiction, and therefore the product is equal to zero.
  • $\mathrm{(D)}$: We are given that $\sum_{k = 1}^{2004}{a_k} = \sum_{k = 1}^{2004}{b_k}$. Since the coefficients are real, it follows that if a root is complex, its conjugate is also a root; and the sum of the imaginary parts of complex conjugates is zero. Hence the RHS is zero.

There is, however, no reason to believe that $\boxed{\mathrm{E}}$ should be zero (in fact, that quantity is $P(1)$, and there is no evidence that $1$ is a root of $P(x)$).

Solution 2(cheap method using answer choices)

Rule out answer choices $A, B, D$ as done in solution 1. Assume for the sake of contradiction that $\mathrm{(E)}$ is $0$. Then $P(1)=0$, so there is some $m\neq 1$ such that $z_m=1+0i$, which implies at least one $b_i$, $i\neq 1$ is $0$ SO if $\sum_{k=1}^{2004}c_k=0$, then $\prod_{i=2}^{2004}b_i=0$. So we have that $\sum_{k=1}^{2004}c_k\neq 0$, because then that would rule out all the answer choices. Hence $\boxed{\mathrm{E}}$ must be nonzero, so the answer is $\mathrm{(E)}$. -vsamc

See also

2004 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 22 		Followed by
Problem 24
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

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