Difference between revisions of "2012 AIME II Problems/Problem 6"
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Revision as of 15:09, 4 July 2013
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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
