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Beastly Polynomial

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December 26, 2006

algebrapolynomialfunction2003 AIME II problem 15

Problem Statement

Let

P(x)=24x^{24}+\sum_{j=1}^{23}(24-j)(x^{24-j}+x^{24+j}).

Let

z_{k}^{2}=a_{k}+b_{k}i

be the distinct zeros of

P(x),

and let

z_{k}^{2}=a_{k}+b_{k}i

for

k=1,2,\ldots,r,

where

i=\sqrt{-1},

and

a_{k}

and

b_{k}

are real numbers. Let

\sum_{k=1}^{r}|b_{k}|=m+n\sqrt{p},

where

m,

n,

and

p

are integers and

p

is not divisible by the square of any prime. Find

m+n+p.

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