Difference between revisions of "2012 AIME II Problems/Problem 6"
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== Solution == | == Solution == | ||
− | {{ | + | Let's consider the maximization constraint first: we want to maximize the value of <math>|z^5 - (1+2i)z^3|</math> |
+ | Simplifying, we have | ||
+ | |||
+ | <math>|z^3| * |z^2 - (1+2i)|</math> | ||
+ | |||
+ | <math>=|z|^3 * |z^2 - (1+2i)|</math> | ||
+ | |||
+ | <math>=125|z^2 - (1+2i)|</math> | ||
+ | |||
+ | Thus we only need to maximize the value of <math>|z^2 - (1+2i)|</math>. | ||
+ | |||
+ | To maximize this value, we must have that <math>z^2</math> is in the opposite direction of <math>1+2i</math>. The unit vector in the complex plane in the desired direction is <math>\frac{-1}{\sqrt{5}} + \frac{-2}{\sqrt{5}} i</math>. Furthermore, we know that the magnitude of <math>z^2</math> is <math>25</math>, because the magnitude of <math>z</math> is <math>5</math>. From this information, we can find that <math>z^2 = \sqrt{5} (-5 - 10i)</math> | ||
+ | |||
+ | Squaring, we get <math>z^4 = 5 (25 - 100 + 100i) = -375 + 500i</math>. Finally, <math>c+d = -375 + 500 = 125</math> | ||
+ | |||
== See Also == | == See Also == | ||
{{AIME box|year=2012|n=II|num-b=5|num-a=7}} | {{AIME box|year=2012|n=II|num-b=5|num-a=7}} |
Revision as of 14:31, 4 April 2012
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,
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 |