59th 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 $d$: $$\begin{array}{rl}\frac{-4}{-4}+\frac{-4}{-2}+\frac{-4}{-1}+\frac{-4}{d}& =\frac{-2}{-4}+\frac{-2}{-2}+\frac{-2}{-1}+\frac{-2}{d}\\ 1+2+4+\frac{-4}{d}& =\frac{1}{2}+1+2+\frac{-2}{d}\\ 7-\frac{4}{d}& =\frac{7}{2}-\frac{2}{d}\\ 7-\frac{4}{d}+\frac{2}{d}& =\frac{7}{2}\\ 7-\frac{2}{d}=\frac{7}{2}& \\ 7-\frac{7}{2}=\frac{2}{d}& \\ \frac{7}{2}=\frac{2}{d}& \\ d=\frac{4}{7}.& \end{array}$$

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