Mock AIME 3 Pre 2005 Problems/Problem 3
Problem
A function is defined for all real numbers
. For all non-zero values
, we have
Let denote the sum of all of the values of
for which
. Compute the integer nearest to
.
Solution
Substituting , we have
This gives us two equations, which we can eliminate
from (the first equation multiplied by two, subtracting the second):
Clearly, the discriminant of the quadratic equation
, so both roots are real. By Vieta's formulas, the sum of the roots is the coefficient of the
term, so our answer is
.
See also
Mock AIME 3 Pre 2005 (Problems, Source) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |