Difference between revisions of "2019 AMC 12B Problems/Problem 17"
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To be equilateral triangle, we should have | To be equilateral triangle, we should have | ||
− | <cmath>\frac{(z^3-z)}{(z-0)}=\text{cis}{(\pi/3)} \text{ or } \text{cis}( | + | <cmath>\frac{(z^3-z)}{(z-0)}=\text{cis}{(\pi/3)} \text{ or } \text{cis}(5\pi/3)</cmath> |
Simplify left side: | Simplify left side: |
Revision as of 22:11, 15 February 2019
Contents
[hide]Problem
How many nonzero complex numbers have the property that
and
when represented by points in the complex plane, are the three distinct vertices of an equilateral triangle?
Solution 1
Convert and
into
form, giving
and
. Since the distance from
to
is
, the distance from
to
must also be
, so
. Now we must find
the requirements for being an equilateral triangle. From
, we have
and from
, we see a monotonic increase of
, from
to
, or equivalently, from
to
. Hence, there are 2 values that work for
. But since the interval
also consists of
going from
to
, it also gives us 2 solutions. Our answer is
Here's a graph of how and
move as
increases- https://www.desmos.com/calculator/xtnpzoqkgs
Someone pls help with LaTeX formatting, thanks -FlatSquare , I did, -Dodgers66
Solution 2
To be equilateral triangle, we should have
Simplify left side:
That is,
We have two roots for both equations, therefore the total number of solution for is
(By Zhen Qin)
See Also
2019 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 16 |
Followed by Problem 18 |
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 |
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions.