Difference between revisions of "Mock AIME 3 Pre 2005 Problems/Problem 3"
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==Problem== | ==Problem== | ||
| − | A function <math>f(x)</math> is defined for all real numbers <math>x</math>. For all non-zero values <math>x</math>, we have | + | A [[function]] <math>f(x)</math> is defined for all real numbers <math>x</math>. For all non-zero values <math>x</math>, we have |
| − | < | + | <cmath>2f\left(x\right) + f\left(\frac{1}{x}\right) = 5x + 4</cmath> |
Let <math>S</math> denote the sum of all of the values of <math>x</math> for which <math>f(x) = 2004</math>. Compute the integer nearest to <math>S</math>. | Let <math>S</math> denote the sum of all of the values of <math>x</math> for which <math>f(x) = 2004</math>. Compute the integer nearest to <math>S</math>. | ||
==Solution== | ==Solution== | ||
| − | {{ | + | Substituting <math>\frac{1}{x}</math>, we have |
| − | ==See | + | <cmath>2f\left(\frac 1x\right) + f\left(x\right) = \frac{5}{x} + 4</cmath> |
| + | |||
| + | This gives us two equations, which we can eliminate <math>f\left(\frac 1x\right)</math> from (the first equation multiplied by two, subtracting the second): | ||
| + | |||
| + | <cmath>\begin{align*} | ||
| + | 3f(x) &= 10x + 4 - \frac 5x \\ | ||
| + | 0 &= x^2 - \frac{3 \times 2004 - 4}{10}x + \frac 52\end{align*}</cmath> | ||
| + | |||
| + | Clearly, the [[discriminant]] of the [[quadratic equation]] <math>\Delta > 0</math>, so both roots are real. By [[Vieta's formulas]], the sum of the roots is the coefficient of the <math>x</math> term, so our answer is <math>\left[\frac{3 \times 2004 - 4}{10}\right] = \boxed{601}</math>. | ||
| + | |||
| + | ==See Also== | ||
| + | {{Mock AIME box|year=Pre 2005|n=3|num-b=2|num-a=4}} | ||
| + | |||
| + | [[Category:Intermediate Algebra Problems]] | ||
Latest revision as of 09:22, 4 April 2012
Problem
A function 
is defined for all real numbers 
. For all non-zero values , we have
![\[2f\left(x\right) + f\left(\frac{1}{x}\right) = 5x + 4\]](http://latex.artofproblemsolving.com/5/8/d/58dd6786465c69dee17d327eff425fb3ee818169.png)
Let 
denote the sum of all of the values of for which 
. Compute the integer nearest to .
Solution
Substituting 
, we have
![\[2f\left(\frac 1x\right) + f\left(x\right) = \frac{5}{x} + 4\]](http://latex.artofproblemsolving.com/b/c/b/bcbf997ecb340d64585b224dc7bf6669fc28d8b4.png)
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 ![$\left[\frac{3 \times 2004 - 4}{10}\right] = \boxed{601}$](http://latex.artofproblemsolving.com/d/3/b/d3b549b1fdac0de532e6832e2cafca5e47466487.png)
.
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 | ||
