Difference between revisions of "1994 AIME Problems/Problem 13"
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== Solution == | == Solution == | ||
− | {{ | + | Let <math>t = 1/x</math>. After multiplying the equation by <math>t^{10}</math>, <math>1 + (13 - t)^{10} = 0\Rightarrow (13 - t)^{10} = - 1</math>. |
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+ | Using DeMoivre, <math>13 - t = e^\frac {(2k + 1)\pi}{10}</math> where <math>k</math> is an integer between <math>0</math> and <math>9</math>. | ||
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+ | <math>t = 13 - e^\frac {(2k + 1)\pi}{10} \Rightarrow \bar{t} = 13 - e^{ - \frac {(2k + 1)\pi}{10}}</math>. | ||
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+ | Since <math>e^{iy} + e^{ - iy} = 2\cos y</math>, <math>t\bar{t} = 170 - 2\cos \frac {(2k + 1)\pi}{10}</math> after expanding. Here <math>k</math> ranges from 0 to 4 because two angles which sum to <math>2\pi</math> are involved in the product.. | ||
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+ | The expression to find is <math>\sum t\bar{t} = 850 - 2\sum_{k = 0}^4 \cos \frac {(2k + 1)\pi}{10}</math>. | ||
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+ | But <math>\cos \frac {\pi}{10} + \cos \frac {9\pi}{10} = \cos \frac {3\pi}{10} + \cos \frac {7\pi}{10} = \cos \frac {\pi}{2} = 0</math> so the sum is <math>\boxed{850}</math>. | ||
== See also == | == See also == |
Revision as of 13:46, 27 November 2008
Problem
The equation
has 10 complex roots where the bar denotes complex conjugation. Find the value of
Solution
Let . After multiplying the equation by , .
Using DeMoivre, where is an integer between and .
.
Since , after expanding. Here ranges from 0 to 4 because two angles which sum to are involved in the product..
The expression to find is .
But so the sum is .
See also
1994 AIME (Problems • Answer Key • Resources) | ||
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 |