2023 WSMO Accuracy Round Problems/Problem 3

Problem

$f(x)=x^3-8x^2+10x-4$ has complex roots $a,b,c$ . Denote $P(n) = a^n+b^n+c^n.$ Find $P(-1)P(0)P(1).$

Solution

We have $\begin{align*} P(-1)P(0)P(1)&=\left(a^{-1}+b^{-1}+c^{-1}\right)\left(a^0+b^0+c^0\right)\left(a^1+b^1+c^1\right)\\ &=\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\left(1+1+1\right)\left(a+b+c\right)\\ &=\frac{3(ab+ac+bc)(a+b+c)}{abc}\\ &=\frac{3(10)(8)}{4} = \boxed{60}. \end{align*}$

~pinkpig

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2023 WSMO Accuracy Round Problems/Problem 3

Problem

Solution