2011 AIME II Problems/Problem 8
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
![2011 AIME II -8.png](https://wiki-images.artofproblemsolving.com//b/b3/2011_AIME_II_-8.png)
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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 |
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