Why Calculus though?

$\displaystyle \sum_{k=1}^{n} x_k^J \le \sum_{k=1}^n \dfrac{1}{x_k^J}$

Given that $J=\frac{3}{2}$ and that $x_1,x_2, \ldots,x_n$ are positive reals whose sum is $n$ , find the largest integer $n$ such that the inequality above always holds true.

If you think all positive integers $n$ make the inequality hold true, enter 0 as your answer.

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Original

Bonus: Solve this question when $J=1.15982$