Difference between revisions of "2012 AIME II Problems/Problem 6"
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== Problem 6 == | == Problem 6 == | ||
Let <math>z=a+bi</math> be the complex number with <math>\vert z \vert = 5</math> and <math>b > 0</math> such that the distance between <math>(1+2i)z^3</math> and <math>z^5</math> is maximized, and let <math>z^4 = c+di</math>. Find <math>c+d</math>. | Let <math>z=a+bi</math> be the complex number with <math>\vert z \vert = 5</math> and <math>b > 0</math> such that the distance between <math>(1+2i)z^3</math> and <math>z^5</math> is maximized, and let <math>z^4 = c+di</math>. Find <math>c+d</math>. |
Revision as of 17:00, 9 August 2018
Contents
Problem 6
Let be the complex number with
and
such that the distance between
and
is maximized, and let
. Find
.
Solution
Let's consider the maximization constraint first: we want to maximize the value of
Simplifying, we have
Thus we only need to maximize the value of .
To maximize this value, we must have that is in the opposite direction of
. The unit vector in the complex plane in the desired direction is
. Furthermore, we know that the magnitude of
is
, because the magnitude of
is
. From this information, we can find that
Squaring, we get . Finally,
Solution 2
WLOG, let and
This means that
Hence, this means that
And
Now, common sense tells us that the distance between these two complex numbers is maxed when they both are points satisfying the equation of the line , or when they are each a
rotation away from each other.
Hence, we must have that
Now, plug this back into (if you want to know why, reread what we want in the problem!)
So now, we have that
Notice that and
Then, we have that and
Finally, plugging back in, we find that
See Also
2012 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.