2010 AIME II Problems/Problem 7
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 |