Difference between revisions of "1991 AIME Problems/Problem 7"

(Solution)
(Solution)
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== Solution ==
 
== Solution ==
The given finite expansion can be easily seen that reduces to solve the quadratic equation <math>x_{}^{2}-\sqrt{19}x-91=0</math>. The solutions are <math>x=\frac{\sqrt{19}\pm\sqrt{383}}{2}</math>. Therefore, <math>A_{}^{}=\vert x_{+}\vert+\vert x_{-}\vert=\sqrt{383}</math>. Therefore, <math>A_{}^{2}=383</math>.
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The given finite expansion can be easily seen that reduces to solve the quadratic equation <math>x_{}^{2}-\sqrt{19}x-91=0</math>. The solutions are <math>\pm</math> <math>x=\frac{\sqrt{19}\pm\sqrt{383}}{2}</math>. Therefore, <math>A_{}^{}=\vert x_{+}\vert+\vert x_{-}\vert=\sqrt{383}</math>. Therefore, <math>A_{}^{2}=383</math>.
  
 
== See also ==
 
== See also ==
 
{{AIME box|year=1991|num-b=6|num-a=8}}
 
{{AIME box|year=1991|num-b=6|num-a=8}}

Revision as of 22:49, 20 April 2007

Problem

Find $A^2_{}$, where $A^{}_{}$ is the sum of the absolute values of all roots of the following equation:

$x = \sqrt{19} + \frac{91}{{\displaystyle \sqrt{19}+\frac{91}{{\displaystyle \sqrt{19}+\frac{91}{{\displaystyle \sqrt{19}+\frac{91}{{\displaystyle \sqrt{19}+\frac{91}{x}}}}}}}}}$

Solution

The given finite expansion can be easily seen that reduces to solve the quadratic equation $x_{}^{2}-\sqrt{19}x-91=0$. The solutions are $\pm$ $x=\frac{\sqrt{19}\pm\sqrt{383}}{2}$. Therefore, $A_{}^{}=\vert x_{+}\vert+\vert x_{-}\vert=\sqrt{383}$. Therefore, $A_{}^{2}=383$.

See also

1991 AIME (ProblemsAnswer KeyResources)
Preceded by
Problem 6
Followed by
Problem 8
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions