Difference between revisions of "Mock AIME 3 Pre 2005 Problems/Problem 4"
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m (Mock AIME 3 Pre 2005/Problem 4 moved to Mock AIME 3 Pre 2005 Problems/Problem 4: naming convention) |
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Revision as of 22:37, 23 April 2008
Problem
and
are complex numbers such that
\zeta_1^2+\zeta_2^2+\zeta_3^2&=3\\
\zeta_1^3+\zeta_2^3+\zeta_3^3&=7\end{align*}$ (Error compiling LaTeX. Unknown error_msg)Compute .
Solution
We let (the elementary symmetric sums). Then, we can rewrite the above equations as
from where it follows that . The third equation can be factored as
from where it follows that . Thus, applying Vieta's formulas backwards,
and
are the roots of the polynomial
Let (the power sums). Then from
, we have the recursion
. It follows that
.
See also
Mock AIME 3 Pre 2005 (Problems, Source) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |