2008 AIME II Problems/Problem 7
Problem
Let , , and be the three roots of the equation Find .
Contents
[hide]Solution
Solution 1
By Vieta's formulas, we have , and so the desired answer is . Additionally, using the factorization we have that . By Vieta's again,
Solution 2
Vieta's formulas gives . Since is a root of the polynomial, , and the same can be done with . Summing these, we have $$ (Error compiling LaTeX. Unknown error_msg)\begin{align*}8\{(r + s)^3 + (s + t)^3 + (t + r)^3\} &= - 8(r^3 + s^3 + t^3)\ &= 1001(r + s + t) + 2008\cdot 3 = 3\cdot 2008$\end{align*}$ (Error compiling LaTeX. Unknown error_msg)753$.
See also
2008 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
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