Difference between revisions of "1991 AIME Problems/Problem 7"
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== Problem == | == Problem == | ||
| − | Find <math>A^2_{}</math>, where <math>A^{}_{}</math> is the sum of the [[absolute value]]s of all roots of the following equation: | + | <!-- don't remove the following tag, for PoTW on the Wiki front page--><onlyinclude>Find <math>A^2_{}</math>, where <math>A^{}_{}</math> is the sum of the [[absolute value]]s of all roots of the following equation: |
<div style="text-align:center"><math>x = \sqrt{19} + \frac{91}{{\sqrt{19}+\frac{91}{{\sqrt{19}+\frac{91}{{\sqrt{19}+\frac{91}{{\sqrt{19}+\frac{91}{x}}}}}}}}} | <div style="text-align:center"><math>x = \sqrt{19} + \frac{91}{{\sqrt{19}+\frac{91}{{\sqrt{19}+\frac{91}{{\sqrt{19}+\frac{91}{{\sqrt{19}+\frac{91}{x}}}}}}}}} | ||
| − | </math></div> | + | </math></div><!-- don't remove the following tag, for PoTW on the Wiki front page--></onlyinclude> |
== Solution == | == Solution == | ||
Revision as of 17:50, 27 March 2015
Problem
Find 
, where 
is the sum of the absolute values of all roots of the following equation:
Solution
Let 
. Then 
, from which we realize that 
. This is because if we expand the entire expression, we will get a fraction of the form 
on the right hand side, which makes the equation simplify to a quadratic. As this quadratic will have two roots, they must be the same roots as the quadratic 
.
The given finite expansion can then be easily seen to reduce to the quadratic equation 
. The solutions are 
. Therefore, 
. We conclude that 
.
See also
| 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
