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

Revision as of 18:22, 4 July 2013

Problem

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

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

Solution

Let $f(x) = \sqrt{19} + \frac{91}{x}$ . Then $x = f(f(f(f(f(x)))))$ , from which we realize that $f(x) = x$ . This is because if we expand the entire expression, we will get a quadratic. As this quadratic will have two roots, they must be the same roots as the quadratic $f(x)=x$ .

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

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	[[Category:Intermediate Algebra Problems]]		[[Category:Intermediate Algebra Problems]]
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1991 AIME (Problems • Answer Key • Resources)
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Difference between revisions of "1991 AIME Problems/Problem 7"

Revision as of 18:22, 4 July 2013

Problem

Solution

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