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 imaginery part of c is 0, since it's the second easiest one to do:
, the imaginery part is:
, which is 0, and therefore x=4, since x=0 don't work,
so now,
and therefore:
, and finally, we have
.