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

Revision as of 21:51, 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 $x_{\pm}^{}$ $=\frac{\sqrt{19}\pm\sqrt{383}}{2}$ . Therefore, $A_{}^{}=\vert x_{+}\vert+\vert x_{-}\vert=\sqrt{383}$ . Therefore, $A_{}^{2}=383$ .

@@ Line 5: / Line 5: @@
 == 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_{\pm}^{}<math></math>=\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>.
+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_{\pm}^{}</math><math>=\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 ==
 {{AIME box|year=1991|num-b=6|num-a=8}}

1991 AIME (Problems • Answer Key • Resources)
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

Page

Toolbox

Search

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

Revision as of 21:51, 20 April 2007

Problem

Solution

See also