I'm trying to prove the following inequality (or something similar, up to a constant factor in either side of the inequality): $$k\cdot\sum_{i=1}^{k}x_{i}\cdot\ln\left(x_{i}\right)\geq\sum_{i=1}^{k}x_{i}\cdot\left(x_{i}-1\right)$$ where $\forall i\in\left[k\right]$, $x_i \in\left[0,k\right]$ (the $x_i$s are not necessarily natural numbers, but we can assume that they're rational if it helps), and $\sum_{i=0}^k x_i=k$.

I've tried plotting it for $k=2,3$ and ran some numerical experiments for larger $k$, and I'm 99% sure this inequality is correct, but I'm still struggling with the proof.

Up to some normalizing, I find the left-hand side quite similar to the entropy of a probability distribution, but I didn't manage to take advantage of this fact either. I also tried looking for inequalities that only hold on simplex-like hyperplanes, but couldn't find anything useful.

Any ideas? Thanks!