2008 iTest Problems/Problem 76

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Problem

During the car ride home, Michael looks back at his recent math exams. A problem on Michael's calculus mid-term gets him starting thinking about a particular quadratic, \[x^2-sx+p,\] with roots $r_1$ and $r_2$. He notices that \[r_1+r_2=r_1^2+r_2^2=r_1^3+r_2^3=\cdots=r_1^{2007}+r_2^{2007}.\] He wonders how often this is the case, and begins exploring other quantities associated with the roots of such a quadratic. He sets out to compute the greatest possible value of \[\dfrac1{r_1^{2008}}+\dfrac1{r_2^{2008}}.\] Help Michael by computing this maximum.

Solution

By Vieta's Formulas, $r_1 + r_2 = s$. That means $r_1^2 + r_2^2 = s^2 - 2p = s$ and $r_1^3 + r_1^3 = (r_1 + r_2)^3 - 3r_1^2r_2 - 3r_1r_2^2 = s^3 - 3ps$.


Note that $s = s^2 - 2p$, so $p = \frac{s^2 - s}{2}$. We also know that $s = s^3 - 3ps$, so substituting for $p$ results in

\begin{align*} s &= s^3 - 3s \cdot \frac{s^2 - s}{2} \\ s &= s^3 - \tfrac32 s^3 + \tfrac32 s^2 \\ 0 &= -\tfrac12 s^3 + \tfrac32 s^2 - s \\ 0 &= s^3 - 3s^2 + 2s \\ &= s(s-2)(s-1) \end{align*}

Thus, $s = 0,1,2$. If $s = 1$ or $s = 0$, then $p = 0$. However, both cases result in one root being zero, so $\dfrac1{r_1^{2008}}+\dfrac1{r_2^{2008}}$ is undefined. If $s = 2$, then $p = 1$, making both roots equal to $1$. Since $1^n = 1$ for $1 \le n \le 2007$, this result satisfies all conditions. Thus, $\dfrac1{r_1^{2008}}+\dfrac1{r_2^{2008}} = 1+1 = \boxed{2}$.

See Also

2008 iTest (Problems)
Preceded by:
Problem 75
Followed by:
Problem 77
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