Difference between revisions of "Mock AIME 2 Pre 2005 Problems/Problem 11"
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<math>\alpha + \beta + \gamma = \frac{799}{4}</math> | <math>\alpha + \beta + \gamma = \frac{799}{4}</math> | ||
| − | <math>\alpha\beta + \beta\gamma + \gamma\alpha | + | |
| + | <math>\alpha\beta + \beta\gamma + \gamma\alpha = -50</math> | ||
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<math>\alpha\beta\gamma = \frac{1}{4}</math> | <math>\alpha\beta\gamma = \frac{1}{4}</math> | ||
Latest revision as of 17:03, 13 September 2020
, 
, and 
are the roots of 
. Let ![\[\omega = \tan^{-1}(\alpha) + \tan^{-1}(\beta) + \tan^{-1} (\gamma).\]](http://latex.artofproblemsolving.com/e/5/6/e561d8bca3fd8b55b65c1b7f6d1f2cb0d911d996.png)
The value of 
can be written as 
where 
and 
are relatively prime positive integers. Determine the value of 
.
Solution
We know that 
are the roots of 
. By Vieta's formulas, we have:
Now, by tangent addition formulas, we have 
. Substituting Vieta's formulas, we obtain 
. Therefore, our answer is 
and we are done.
See also
| Mock AIME 2 Pre 2005 (Problems, Source) | ||
| Preceded by Problem 10 |
Followed by Problem 12 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
