Difference between revisions of "2011 AIME II Problems/Problem 8"
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== Problem == | == Problem == | ||
| − | Let <math>z_1 | + | Let <math>z_1,z_2,z_3,\dots,z_{12}</math> be the 12 zeroes of the polynomial <math>z^{12}-2^{36}</math>. For each <math>j</math>, let <math>w_j</math> be one of <math>z_j</math> or <math>i z_j</math>. Then the maximum possible value of the real part of <math>\displaystyle\sum_{j=1}^{12} w_j</math> can be written as <math>m+\sqrt{n}</math> where <math>m</math> and <math>n</math> are positive integers. Find <math>m+n</math>. |
== Solution == | == Solution == | ||
Revision as of 21:57, 17 March 2019
Problem
Let 
be the 12 zeroes of the polynomial 
. For each 
, let 
be one of 
or 
. Then the maximum possible value of the real part of 
can be written as 
where 
and 
are positive integers. Find 
.
Solution
The twelve dots above represent the 12 roots of the equation 
. If we write 
, then the real part of 
is 
and the real part of 
is 
. The blue dots represent those roots for which the real part of is greater than the real part of , and the red dots represent those roots for which the real part of is greater than the real part of . Now, the sum of the real parts of the blue dots is easily seen to be 
and the negative of the sum of the imaginary parts of the red dots is easily seen to also be 
. Hence our desired sum is 
, giving the answer 
.
See also
| 2011 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 7 |
Followed by Problem 9 | |
| 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. 
