Difference between revisions of "1995 AIME Problems/Problem 5"
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Revision as of 18:30, 4 July 2013
Problem
For certain real values of 
and 
the equation 
has four non-real roots. The product of two of these roots is 
and the sum of the other two roots is 
where 
Find 
Solution
Since the coefficients of the polynomial are real, it follows that the non-real roots must come in complex conjugate pairs. Let the first two roots be 
. Since 
is not real, are not conjugates, so the other pair of roots must be the conjugates of . Let 
be the conjugate of 
, and 
be the conjugate of 
. Then,
![\[m\cdot n = 13 + i,m' + n' = 3 + 4i\Longrightarrow m'\cdot n' = 13 - i,m + n = 3 - 4i.\]](http://latex.artofproblemsolving.com/d/a/b/dab8020a8532d068d90e04c999b3b3959a13e140.png)
By Vieta's formulas, we have that 
.
See also
| 1995 AIME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 4 |
Followed by Problem 6 | |
| 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. 
