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Difference between revisions of "2023 WSMO Accuracy Round Problems/Problem 3"

(Created page with "==Problem== <math>f(x)=x^3-8x^2+10x-4</math> has complex roots <math>a,b,c</math>. Denote <math>P(n) = a^n+b^n+c^n.</math> Find <math>P(-1)P(0)P(1).</math> ==Solution==")
 
 
Line 4: Line 4:
  
 
==Solution==
 
==Solution==
 +
We have
 +
<cmath>\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*}</cmath>
 +
 +
~pinkpig

Latest revision as of 11:43, 13 September 2025

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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