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Difference between revisions of "Mock AIME 3 2006-2007/Problem 3"

(Created page with '== Problem == Let <math>P(x) = 2x^{59} - x^2 - x - 6</math>. If <math>Q(x)</math> is a polynomial whose roots are the 59th powers of the roots of <math>P(x)</math>, then find th…')
 
(No difference)

Latest revision as of 00:35, 30 December 2009

Problem

Let $P(x) = 2x^{59} - x^2 - x - 6$. If $Q(x)$ is a polynomial whose roots are the 59th powers of the roots of $P(x)$, then find the sum of the roots of $Q(x)$.

Solution

Let $a_1,a_2,\ldots,a_{59}$ denote the roots of $P$. Then, $\sum_{i} a_i = 0$, and $\sum_{i\neq j} = 0$. Therefore, $\sum_{i} a_i^2 = 0$.

Denote $S_n(a) = \sum_{i} a_i^n$. So $S_1(a) = 0, S_2(a) = 0$. To find $S_{59}(a)$, use Newton's recursive sums:

\[S_{59}(a)\cdot 2 + S_{58}(a)\cdot 0 + S_{57}(a)\cdot 0 + \cdots + S_2(a)\cdot(-1) + S_1(a)\cdot(-1) + 59\cdot(-6) = 0.\]

Solving, we get $S_{59}(a) = 59\cdot 3 = \boxed{177}$.

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