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1991 AIME Problems/Problem 7

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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 hypothesize that 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.

See also

1991 AIME (ProblemsResources)
Preceded by
Problem 6
Followed by
Problem 8
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Looking for a challenging algebra text? Preparing for MATHCOUNTS or the AMC exams?
Check out Art of Problem Solving's Introduction to Algebra by Richard Rusczyk.
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