MHB Can You Prove Inequality Challenge II?

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The cubic polynomial \( p(x) = x^3 + mx^2 + nx + k \) has three distinct real roots, while the transformed polynomial \( (x^2 + x + 2014)^3 + m(x^2 + x + 2014)^2 + n(x^2 + x + 2014) + k = 0 \) has no real roots. This leads to the conclusion that the discriminant of \( x^2 + x + 2014 - \alpha \) must be negative for each root \( \alpha, \beta, \gamma \), resulting in \( 2014 - \alpha > \frac{1}{4} \). Consequently, it follows that \( k + 2014n + 2014^2m + 2014^3 > \frac{1}{64} \). The proof hinges on the relationships between the roots and the conditions imposed by the lack of real roots in the transformed polynomial.
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The cubic polynomial $x^3+mx^2+nx+k=0$ has three distinct real roots but the other polynomial $(x^2+x+2014)^3+m(x^2+x+2014)^2+n(x^2+x+2014)+k=0$ has no real roots. Show that $k+2014n+2014^2m+2014^3>\dfrac{1}{64}$.
 
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[sp]Let $p(x) = x^3+mx^2+nx+k$, and let $\alpha,\beta,\gamma$ be its three real roots, so that $p(x) = (x-\alpha)(x - \beta)(x - \gamma).$ Let $Y = 2014.$ If $p(x^2 + x + Y) = 0$ then $x^2+x+Y$ must be a root of $p(x)$, say $x^2+x+Y = \alpha.$ But if $p(x^2 + x + Y) = 0$ has no real roots then $x^2+x+ Y - \alpha = 0$ must have no real roots, so its discriminant must be negative: $1 - 4(Y- \alpha) < 0.$ Therefore $Y-\alpha > \frac14$, and similarly $Y-\beta > \frac14$, $Y-\gamma > \frac14$. It follows that $$ k+2014n+2014^2m+2014^3 = p(Y) = (Y -\alpha)(Y - \beta)(Y - \gamma) > \tfrac14\cdot \tfrac14\cdot \tfrac14 = \tfrac1{64}.$$[/sp]
 
Opalg said:
[sp]Let $p(x) = x^3+mx^2+nx+k$, and let $\alpha,\beta,\gamma$ be its three real roots, so that $p(x) = (x-\alpha)(x - \beta)(x - \gamma).$ Let $Y = 2014.$ If $p(x^2 + x + Y) = 0$ then $x^2+x+Y$ must be a root of $p(x)$, say $x^2+x+Y = \alpha.$ But if $p(x^2 + x + Y) = 0$ has no real roots then $x^2+x+ Y - \alpha = 0$ must have no real roots, so its discriminant must be negative: $1 - 4(Y- \alpha) < 0.$ Therefore $Y-\alpha > \frac14$, and similarly $Y-\beta > \frac14$, $Y-\gamma > \frac14$. It follows that $$ k+2014n+2014^2m+2014^3 = p(Y) = (Y -\alpha)(Y - \beta)(Y - \gamma) > \tfrac14\cdot \tfrac14\cdot \tfrac14 = \tfrac1{64}.$$[/sp]

Bravo, Opalg!...and thanks for participating!:)
 
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