2001 AIME II Problems/Problem 14
Problem
There are 
complex numbers that satisfy both 
and 
. These numbers have the form 
, where 
and angles are measured in degrees. Find the value of 
.
Solution
Z can be written in the form 
. Rearranging, we get 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 One :
= 
and 
= 
and 
Setting up and solving equations, $Z^{28}= cis{60^\circ$ (Error compiling LaTeX. Unknown error_msg) and $Z^8= cis{120^\circ$ (Error compiling LaTeX. Unknown error_msg), we see that the solutions common to both equations have arguments 
and 
- Case 2 : =
and = 
and 
Again setting up equations ($Z^{28}= cis{300^\circ$ (Error compiling LaTeX. Unknown error_msg) and $Z^{8} = cis{240^\circ$ (Error compiling LaTeX. Unknown error_msg)) we see that the common solutions have arguments of 
and 
Listing all of these values, it is seen that is equal to 
which is equal to 
degrees
See also
| 2001 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 13 |
Followed by Problem 15 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
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