1994 AIME Problems/Problem 13
Contents
Problem
The equation
has 10 complex roots where the bar denotes complex conjugation. Find the value of
Solution 1
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 .
Solution 2
Divide both sides by to get
Rearranging:
Thus, where where is an integer.
We see that . Thus,
Summing over all terms:
However, note that from drawing the numbers on the complex plane, our answer is just
See also
1994 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
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