AoPSWiki
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.

1991 AIME Problems/Problem 7

From AoPSWiki

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 (Problems • Resources)
Preceded by
Problem 6 		Followed by
Problem 8
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
Retrieved from "http://www.artofproblemsolving.com/Wiki/index.php/1991_AIME_Problems/Problem_7"
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.
© Copyright 2008 AoPS Incorporated. All Rights Reserved. • Foundation • Privacy • Contact Us