MHB Can the Inequality $x,y,z>1$ be Proven with a Hint of 48?

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The discussion revolves around proving the inequality involving positive variables x, y, and z, each greater than 1. The specific inequality to be proven is that the sum of the fractions, $\dfrac{x^4}{(y-1)^2}+\dfrac{y^4}{(z-1)^2}+\dfrac{z^4}{(x-1)^2}$, is greater than or equal to 48. Participants are encouraged to explore the hint provided, which suggests a potential method or insight for the proof. The focus remains on finding a rigorous mathematical approach to establish the inequality under the given conditions. The conversation emphasizes the need for clarity in the proof process and the importance of the hint in guiding the solution.
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$x,y,z>1$

please prove :

$\dfrac{x^4}{(y-1)^2}+\dfrac{y^4}{(z-1)^2}+\dfrac{z^4}{(x-1)^2}\geq 48$
 
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Albert said:
$x,y,z>1$

please prove :

$\dfrac{x^4}{(y-1)^2}+\dfrac{y^4}{(z-1)^2}+\dfrac{z^4}{(x-1)^2}\geq 48$
hint:
prove :$\dfrac{x^4}{(y-1)^2}\geq 32(x-y)+16$
 
Albert said:
hint:
prove :$\dfrac{x^4}{(y-1)^2}\geq 32(x-y)+16$
$\dfrac{x^4}{(y-1)^2}+16(y-1)+16(y-1)+16\geq 4\sqrt[4]{16^3x^4}= 32x$
or :$\dfrac{x^4}{(y-1)^2}\geq 32(x-y)+16$
please complete the rest
 

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