Difference between revisions of "2003 AIME II Problems/Problem 15"

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== Problem ==
 
== Problem ==
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Let
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<center><math>P(x) = 24x^{24} + \sum_{j = 1}^{23}(24 - j)(x^{24 - j} + x^{24 + j}).</math></center>
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Let <math>z_{1},z_{2},\ldots,z_{r}</math> be the distinct zeros of <math>P(x),</math> and let <math>z_{K}^{2} = a_{k} + b_{k}i</math> for <math>k = 1,2,\ldots,r,</math> where <math>i = \sqrt { - 1},</math> and <math>a_{k}</math> and <math>b_{k}</math> are real numbers. Let
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<center><math>\sum_{k = 1}^{r}|b_{k}| = m + n\sqrt {p},</math></center>
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where <math>m,</math> <math>n,</math> and <math>p</math> are integers and <math>p</math> is not divisible by the square of any prime. Find <math>m + n + p.</math>
  
 
== Solution ==
 
== Solution ==
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== See also ==
 
== See also ==
* [[2003 AIME II Problems/Problem 14| Previous problem]]
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{{AIME box|year=2003|n=II|num-b=14|after=Last Question}}
 
 
* [[2003 AIME II Problems]]
 

Revision as of 14:42, 21 November 2007

Problem

Let

$P(x) = 24x^{24} + \sum_{j = 1}^{23}(24 - j)(x^{24 - j} + x^{24 + j}).$

Let $z_{1},z_{2},\ldots,z_{r}$ be the distinct zeros of $P(x),$ and let $z_{K}^{2} = a_{k} + b_{k}i$ for $k = 1,2,\ldots,r,$ where $i = \sqrt { - 1},$ and $a_{k}$ and $b_{k}$ are real numbers. Let

$\sum_{k = 1}^{r}|b_{k}| = m + n\sqrt {p},$

where $m,$ $n,$ and $p$ are integers and $p$ is not divisible by the square of any prime. Find $m + n + p.$

Solution

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See also

2003 AIME II (ProblemsAnswer KeyResources)
Preceded by
Problem 14
Followed by
Last Question
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All AIME Problems and Solutions
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