This problem has to be the real and integral part of Christmas... :)
Here goes:
What is the minimum value of $k$ such that $$\Re\left\{\int_{e^{\large 1/\pi}}^{\pi^{\large 1/e}}\left[% \left(\frac{x^e}{\pi^{x}}\right)^{ex/\pi} - \left(\frac{e^{x}}{x^{\pi}}\right)^{\pi x/e}\right]^{k}\,\mathrm{d}x\right\} = 1? $$
edit: @DonnyFrank has provided a visualization showing that $k$ can take infinitely many values
My Observations
> The integral is negative for odd integers of $k$ and positive for even integers of $k$.
> The integral is complex for non-integer $k$ (it is already complex for any value of $k$ ;)
> The real part of the integral is very close to $1$ as $k$ approaches $10$ - in fact it surpasses $10$ at around $k=9.897$.
My Attempt
No clue. I know that $k$ is not an integer so the binomial theorem wouldn't help at all.
Merry Christmas and a Happy New Year!