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 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |