Difference between revisions of "2020 AMC 12B Problems/Problem 23"
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For <math>n=2</math>, we see that if <math>z_{1}+z_{2}=0</math>, then <math>z_{1}=-z_{2}</math>, so they are evenly spaced along the unit circle. | For <math>n=2</math>, we see that if <math>z_{1}+z_{2}=0</math>, then <math>z_{1}=-z_{2}</math>, so they are evenly spaced along the unit circle. | ||
Revision as of 10:55, 20 June 2022
Contents
[hide]Problem
How many integers are there such that whenever
are complex numbers such that
then the numbers
are equally spaced on the unit circle in the complex plane?
Solutions
For , we see that if
, then
, so they are evenly spaced along the unit circle.
For , WLOG, we can set
. Notice that now
and
. This forces
and
to be equal to
and
, meaning that all three are equally spaced along the unit circle.
We can now show that we can construct complex numbers when that do not satisfy the conditions in the problem.
Suppose that the condition in the problem holds for some . We can now add two points
and
anywhere on the unit circle such that
, which will break the condition. Now that we have shown that
and
works, by this construction, any
does not work, making the answer
.
-Solution by Qqqwerw
Video Solution
On The Spot STEM: https://www.youtube.com/watch?v=JOgSOni5HhM
See Also
2020 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 22 |
Followed by Problem 24 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.