2019 AMC 12B Problems/Problem 17
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
Convert and
into
form, giving
and
. Since the distance from
to
is
, the distance from
to
must also be
, so
. Now we must find
. From
, we have
and from
, we see a monotonic decrease of
, from
to
. Hence, there are 2 values that work for
. But since the interval
is identical, because
at pi, we have 4 solutions. There are not infinitely many solutions since the same four solutions are duplicated.
Here's a graph of how the points move as
\[\theta\] (Error making remote request. Unexpected URL sent back)
increases- https://www.desmos.com/calculator/xtnpzoqkgs
Someone pls help with LaTeX formatting, thanks -FlatSquare , I did, -Dodgers66
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 |