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Difference between revisions of "2024 SSMO Team Round Problems/Problem 6"

(Created page with "==Problem== Let <math>\alpha</math>, <math>\beta</math>, and <math>\gamma</math> be the roots of the polynomial <math>x^3 - 6x^2 - 19x - n</math>. If <math>n</math> is an int...")
 
 
Line 4: Line 4:
  
 
==Solution==
 
==Solution==
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From Vieta's Formulas, we have <cmath>\alpha+\beta+\gamma = 6,\alpha\beta+\alpha\gamma+\beta\gamma = -19,\text{ and }\alpha\beta\gamma = n.</cmath> Since <math>x^3-6x^2-19x-n = 0</math> for <math>x \in \{\alpha,\beta,\gamma\},</math> we have <math>x^3 = 6x^2+19x+n.</math> So,
 +
\begin{align*}
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\alpha^3+\beta^3+\gamma^3&=(6\alpha^2+19\alpha+n)+(6\beta^2+19\beta+n)\\&+(6\gamma^2+19\gamma+n)\\
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&=6(\alpha^2+\beta^2+\gamma^2)+19(\alpha+\beta+\gamma)+3n\\
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&=6((\alpha+\beta+\gamma)^2-2(\alpha\beta+\alpha\gamma+\beta\gamma))\\
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&+19(\alpha+\beta+\gamma)+3n\\
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&=6((6^2)-2(-19))+19(6)+3n = 558+3n.
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\end{align*}
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Since <math>n</math> is an integer and we are seeking to find the least positive value of <math>558+3n = 3(n+186),</math> we let <math>n = -185,</math> giving an answer of <math>\boxed{3}.</math>
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 +
~SMO_Team

Latest revision as of 14:37, 10 September 2025

Problem

Let $\alpha$, $\beta$, and $\gamma$ be the roots of the polynomial $x^3 - 6x^2 - 19x - n$. If $n$ is an integer, what is the least possible positive value of $\alpha^3 + \beta^3 + \gamma^3$?

Solution

From Vieta's Formulas, we have \[\alpha+\beta+\gamma = 6,\alpha\beta+\alpha\gamma+\beta\gamma = -19,\text{ and }\alpha\beta\gamma = n.\] Since $x^3-6x^2-19x-n = 0$ for $x \in \{\alpha,\beta,\gamma\},$ we have $x^3 = 6x^2+19x+n.$ So, \begin{align*} \alpha^3+\beta^3+\gamma^3&=(6\alpha^2+19\alpha+n)+(6\beta^2+19\beta+n)\\&+(6\gamma^2+19\gamma+n)\\ &=6(\alpha^2+\beta^2+\gamma^2)+19(\alpha+\beta+\gamma)+3n\\ &=6((\alpha+\beta+\gamma)^2-2(\alpha\beta+\alpha\gamma+\beta\gamma))\\ &+19(\alpha+\beta+\gamma)+3n\\ &=6((6^2)-2(-19))+19(6)+3n = 558+3n. \end{align*} Since $n$ is an integer and we are seeking to find the least positive value of $558+3n = 3(n+186),$ we let $n = -185,$ giving an answer of $\boxed{3}.$

~SMO_Team

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