Difference between revisions of "2010 AIME II Problems/Problem 7"
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Since <math>a,b,c\in{R}</math>, the imaginary part of a,b,c must be 0. | Since <math>a,b,c\in{R}</math>, the imaginary part of a,b,c must be 0. | ||
− | Start with a, since it's the easiest one to do: <math>y+3+y+9+2y=0, y=-3</math> | + | Start with a, since it's the easiest one to do: <math>y+3+y+9+2y=0, y=-3</math>, |
− | and therefore: <math>x_1 = x</math>, <math>x_2 = x+6i</math>, <math>x_3 = 2x-4-6i</math> | + | and therefore: <math>x_1 = x</math>, <math>x_2 = x+6i</math>, <math>x_3 = 2x-4-6i</math>. |
Now, do the part where the imaginary part of c is 0, since it's the second easiest one to do: | Now, do the part where the imaginary part of c is 0, since it's the second easiest one to do: | ||
− | <math>x(x+6i)(2x-4-6i)</math> | + | <math>x(x+6i)(2x-4-6i)</math>. The imaginary part is: <math>6x^2-24x</math>, which is 0, and therefore x=4, since x=0 doesn't work. |
− | So now, <math>x_1 = 4, x_2 = 4+6i, x_3 = 4-6i</math> | + | So now, <math>x_1 = 4, x_2 = 4+6i, x_3 = 4-6i</math>, |
− | and therefore: <math>a=-12, b=84, c=-208</math> | + | and therefore: <math>a=-12, b=84, c=-208</math>. Finally, we have <math>|a+b+c|=|-12+84-208|=\boxed{136}</math>. |
== See also == | == See also == | ||
{{AIME box|year=2010|num-b=6|num-a=8|n=II}} | {{AIME box|year=2010|num-b=6|num-a=8|n=II}} |
Revision as of 22:56, 5 March 2012
Problem 7
Let , where a, b, and c are real. There exists a complex number such that the three roots of are , , and , where . Find .
Solution
Set , so , , .
Since , the imaginary part of a,b,c must be 0.
Start with a, since it's the easiest one to do: ,
and therefore: , , .
Now, do the part where the imaginary part of c is 0, since it's the second easiest one to do: . The imaginary part is: , which is 0, and therefore x=4, since x=0 doesn't work.
So now, ,
and therefore: . Finally, we have .
See also
2010 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |