Sum of rounded down numbers

Question

I have a sequence of numbers $x_1, x_2, ... x_n$ that I know add up to $k$. I want to show that $\sum_{i=0}\lfloor{\frac{x_i}{d}}\rfloor = \Omega(\frac{k}{d})$. Is this even true? If so - how would one go about showing it? If not, how would one go about evaluating this sum properly and obtaining a proper lower bound?