Mock AIME 3 2006-2007 Problems/Problem 3

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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Mock AIME 3 2006-2007 Problems/Problem 3

Problem

Solution