2011 AIME II Problems/Problem 8
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 .
The twelve dots above represent the 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
As a small note, we could factor the equation as in which these are just the 12th roots of unity for which might be easier to work with. Then we just add up the optimal real parts as in solution 1 and then multiply by 8 at the end.
Note that .
So, one can easily translate all the red dots by .
Then, it's easy to see that points 2, 3, and 4 cancel out with 5, 6, and 7, so you're left with times two
The equation can be factored as follows:
Since this is a 12th degree equation, there are 12 roots. Also, since each term in the equation is even, the positive or negative value of each root is another root. That would mean there are 6 roots that can be multiplied by and since we have 6 factors, that’s 1 root per factor. We just need to solve for in each factor and pick whether or not to multiply by and for each one depending on the one that yields the highest real value. After that process, we get Adding the values up yields , or , and .
-Solution by Someonenumber011.
Clearly, the roots are: .
Now, realize for , . . is true when .
When , .
When , .
For the 12 roots of the polynomial in the original equation, . .
So . . This can be easily true for roots that are in the 3rd quadrant in the complex plane.
This cannot be true for roots in the 1st quadrant because that would yield a negative number bigger than a positive one.
Consider the roots in the 2nd and 4th quadrants. Calculate the roots, choose, and then add the ones up. You will get .
Solution 4 (Brute Force)
Use De Moivre's Theorem to brute force all the roots out. Then choose the greater value of . After adeding everything up, you get .
|2011 AIME II (Problems • Answer Key • Resources)|
|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.