2001 AIME II Problems/Problem 14
There are complex numbers that satisfy both and . These numbers have the form , where and angles are measured in degrees. Find the value of .
can be written in the form . Rearranging, we find that
Since the real part of is one more than the real part of and their imaginary parts are equal, it is clear that either and , or and
- Case 1 : and
Setting up and solving equations, and , we see that the solutions common to both equations have arguments and . We can figure this out by adding 360 repeatedly to the number 60 to try and see if it will satisfy what we need. We realize that it does not work in the integer values.
- Case 2 : and
Again setting up equations ( and ) we see that the common solutions have arguments of and
Listing all of these values, we find that is equal to which is equal to degrees. We only want the sum of a certain number of theta, not all of it.
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