Difference between revisions of "1994 AIME Problems/Problem 13"

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The equation
 
The equation
 
<center><math>x^{10}+(13x-1)^{10}=0\,</math></center>
 
<center><math>x^{10}+(13x-1)^{10}=0\,</math></center>
has 10 complex roots <math>r_1, \overline{r_1}, r_2, \overline{r_2}, r_3, \overline{r_3}, r_4, \overline{r_4}, r_5, \overline{r_5},\,</math> where the bar denotes complex conjugation.  Find the value of
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has 10 [[complex number|complex]] [[root]]s <math>r_1, \overline{r_1}, r_2, \overline{r_2}, r_3, \overline{r_3}, r_4, \overline{r_4}, r_5, \overline{r_5},\,</math> where the bar denotes complex conjugation.  Find the value of
 
<center><math>\frac 1{r_1\overline{r_1}}+\frac 1{r_2\overline{r_2}}+\frac 1{r_3\overline{r_3}}+\frac 1{r_4\overline{r_4}}+\frac 1{r_5\overline{r_5}}.</math></center>
 
<center><math>\frac 1{r_1\overline{r_1}}+\frac 1{r_2\overline{r_2}}+\frac 1{r_3\overline{r_3}}+\frac 1{r_4\overline{r_4}}+\frac 1{r_5\overline{r_5}}.</math></center>
  
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== See also ==
 
== See also ==
 
{{AIME box|year=1994|num-b=12|num-a=14}}
 
{{AIME box|year=1994|num-b=12|num-a=14}}
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[[Category:Intermediate Algebra Problems]]

Revision as of 19:02, 4 December 2007

Problem

The equation

$x^{10}+(13x-1)^{10}=0\,$

has 10 complex roots $r_1, \overline{r_1}, r_2, \overline{r_2}, r_3, \overline{r_3}, r_4, \overline{r_4}, r_5, \overline{r_5},\,$ where the bar denotes complex conjugation. Find the value of

$\frac 1{r_1\overline{r_1}}+\frac 1{r_2\overline{r_2}}+\frac 1{r_3\overline{r_3}}+\frac 1{r_4\overline{r_4}}+\frac 1{r_5\overline{r_5}}.$

Solution

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See also

1994 AIME (ProblemsAnswer KeyResources)
Preceded by
Problem 12
Followed by
Problem 14
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions
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