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. | ||
− | For <math>n=3</math>, WLOG, we can set <math>z_{1}=1</math>. Notice that now <math>\Re(z_{2}+z{3})=-1</math> and <math>\Im\{z_{2}\}=-\Im\{z_{3}\}</math>. This forces <math>z_{2}</math> and <math>z_{3}</math> to be equal to <math>e^{i\frac{\pi}{3}}</math> and <math>e^{-i\frac{\pi}{3}}</math> | + | For <math>n=3</math>, WLOG, we can set <math>z_{1}=1</math>. Notice that now <math>\Re(z_{2}+z{3})=-1</math> and <math>\Im\{z_{2}\}=-\Im\{z_{3}\}</math>. This forces <math>z_{2}</math> and <math>z_{3}</math> to be equal to <math>e^{i\frac{2\pi}{3}}</math> and <math>e^{-i\frac{2\pi}{3}}</math>, meaning that all three are equally spaced along the unit circle. |
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+ | We can now show that we can construct complex numbers when <math>n\geq 4</math> that do not satisfy the conditions in the problem. | ||
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+ | Consider <math>n=4</math>. We can set <math>z_{1}</math> and <math>z_{3}</math> arbitrarily and let <math>z_{2}=-z_{1}</math> and <math>z_{4}=-z_{3}</math>. We constructed two pairs of points that cancel each other out. For example, the complex numbers <math>1,-1,e^{i\frac{\pi}{4}},e^{-i\frac{3\pi}{4}}</math> | ||
==See Also== | ==See Also== |
Revision as of 00:03, 8 February 2020
Problem 23
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?
Solution
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.
Consider . We can set
and
arbitrarily and let
and
. We constructed two pairs of points that cancel each other out. For example, the complex numbers
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 |
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