Difference between revisions of "2008 AIME II Problems/Problem 7"
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Find <math>(r + s)^3 + (s + t)^3 + (t + r)^3</math>. | Find <math>(r + s)^3 + (s + t)^3 + (t + r)^3</math>. | ||
+ | __TOC__ | ||
== Solution == | == Solution == | ||
− | By [[ | + | === Solution 1 === |
+ | By [[Vieta's formulas]], we have <math>r+s+t = 0</math>, and so the desired answer is <math>(r+s)^3 + (s+t)^3 + (t+r)^3 = (0-t)^3 + (0-r)^3 + (0-s)^3 = -(r^3 + s^3 + t^3)</math>. Additionally, using the factorization | ||
<cmath>r^3 + s^3 + t^3 - 3rst = (r+s+t)(r^2 + s^2 + t^2 - rs - st - tr) = 0</cmath> | <cmath>r^3 + s^3 + t^3 - 3rst = (r+s+t)(r^2 + s^2 + t^2 - rs - st - tr) = 0</cmath> | ||
− | + | we have that <math>r^3 + s^3 + t^3 = 3rst</math>. By Vieta's again, <math>rst = \frac{-2008}8 = -251 \Longrightarrow -(r^3 + s^3 + t^3) = -3rst = \boxed{753}.</math> | |
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+ | === Solution 2 === | ||
+ | Vieta's formulas gives <math>r + s + t = 0</math>. Since <math>r</math> is a root of the polynomial, <math>8r^3 + 1001r + 2008 = 0\Longleftrightarrow - 8r^3 = 1001r + 2008</math>, and the same can be done with <math>s,\ t</math>. Summing these, we have | ||
+ | <math></math>\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<math>\end{align*}</math><math>, yielding the answer </math>753$. | ||
== See also == | == See also == |
Revision as of 16:56, 3 April 2008
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 |