Difference between revisions of "Mock AIME 2 Pre 2005 Problems/Problem 11"
Duck master (talk | contribs) (created page w/ solution & categorization) |
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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> |
<math>\alpha\beta\gamma = \frac{1}{4}</math> | <math>\alpha\beta\gamma = \frac{1}{4}</math> | ||
Revision as of 11:49, 19 August 2020
,
, and
are the roots of
. Let
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 |