What is the minimum value of a complex expression?

[Ackbach]

Here is this week's POTW:

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Find the minimum value of
\[\frac{(x+1/x)^6-(x^6+1/x^6)-2}{(x+1/x)^3+(x^3+1/x^3)}\]
for $x>0$.

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[Ackbach]

Re: Problem Of The Week # 244 - Dec 12, 2016

This was Problem B-1 in the 1998 William Lowell Putnam Mathematical Competition.

Congratulations to Opalg for his correct solution, which follows:

Spoiler

In the denominator, $(x+1/x)^3+(x^3+1/x^3) = 2x^3 + 3x + 3/x + 2/x^3 = 2(x+1/x)^3 - 3(x+1/x).$

In the numerator, $(x+1/x)^6-(x^6+1/x^6)-2 = 6x^4 + 15x^2 + 18 + 15/x^2 + 6/x^4 = 6(x+1/x)^4 - 9(x+1/x)^2.$

Put those together and let $y= x+1/x$, getting $$\frac{(x+1/x)^6-(x^6+1/x^6)-2}{(x+1/x)^3+(x^3+1/x^3)} = \frac{6y^4 - 9y^2}{2y^3 - 3y} = \frac{3y(2y^3 - 3y)}{2y^3 - 3y} = 3y = 3(x+1/x).$$ The minimum value of $x+1/x$ is $2$, at $x=1$. Therefore the minimum value of $\frac{(x+1/x)^6-(x^6+1/x^6)-2}{(x+1/x)^3+(x^3+1/x^3)}$ is $6$.