Ukrainian National Mathematical Olympiad

Answer. Yes, for example $a = -28$, $b = -14$, $c = -7$ and $d = 4$.

Clearly, all numbers cannot be positive. Let's take $a = -4$, $b = -2$, $c = -1$ and find the corresponding $d$:
$$\begin{aligned} \frac{-4}{-4} + \frac{-4}{-2} + \frac{-4}{-1} + \frac{-4}{d} &= \frac{-2}{-4} + \frac{-2}{-2} + \frac{-2}{-1} + \frac{-2}{d} \\ 1 + 2 + 4 + \frac{-4}{d} &= \frac{1}{2} + 1 + 2 + \frac{-2}{d} \\ 7 - \frac{4}{d} &= \frac{7}{2} - \frac{2}{d} \\ 7 - \frac{4}{d} + \frac{2}{d} &= \frac{7}{2} \\ 7 - \frac{2}{d} = \frac{7}{2} \\ 7 - \frac{7}{2} = \frac{2}{d} \\ \frac{7}{2} = \frac{2}{d} \\ d = \frac{4}{7}. \end{aligned}$$

The equality is homogeneous, meaning that we can simply multiply all numbers by $7$ to obtain the aforementioned integers.