2005 AIME I Problems/Problem 8
From AoPSWiki
Problem
The equation 
has three real roots. Given that their sum is 
where 
and 
are relatively prime positive integers, find 
Solution
Let 
. Then our equation reads 
or 
. Thus, if this equation has roots 
and 
, by Vieta's formulas we have 
. Let the corresponding values of 
be 
and 
. Then the previous statement says that 
so that taking a logarithm gives 
and 
. Thus the answer is 
.
See also
| 2005 AIME I (Problems • Resources) | ||
| Preceded by Problem 7 | Followed by Problem 9 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
