19. Let the number $a$ have the following property: For any four real numbers $x_{1}, x_{2}, x_{3}, x_{4}$, there always exist integers $k_{1}, k_{2}, k_{3}, k_{4}$, such that $\sum_{1 \leqslant i<j \leqslant 4}\left(\left(x_{i}-k_{i}\right)-\left(x_{j}-k_{j}\right)\right)^{2} \leqslant a$, find the minimum value of such $a$. (1990
19. First, let $k_{i}=\left[x_{i}\right], \alpha_{i}=x_{i}-k_{i}$, then $\alpha_{i} \in[0,1), i=1,2,3,4$. Let $\left\{\alpha^{\prime}{ }_{1}, \alpha^{\prime}{ }_{2}, \alpha^{\prime}{ }_{3}, \alpha^{\prime}{ }_{4}\right\}$ be a permutation of $\left\{\alpha_{1}, \alpha_{2}, \alpha_{3}, \alpha_{4}\right\}$ in ascending order, and denote $\beta_{1}=\alpha^{\prime}{ }_{2}-\alpha_{1}^{\prime}$, $\beta_{2}=\alpha_{3}^{\prime}-\alpha_{2}^{\prime}, \beta_{3}=\alpha_{4}^{\prime}-\alpha_{3}^{\prime}, \beta_{4}=1-\alpha_{4}^{\prime}+\alpha_{1}^{\prime}$, then,
$$\begin{aligned}
S= & \sum_{1 \leqslant i0$ when, $\beta_{1}$ and $\beta_{3}$ each decrease by $\frac{1}{2}$ of the increment of $\beta_{2}$; when $\beta_{3}=0$, $\beta_{3}$ decreases by the increment of $\beta_{2}$. In either case, the value of the expression for $\dot{S}$ must increase, i.e.,
$$S \leqslant 2\left[\beta_{4}^{2}+\left(1-\beta_{4}\right)^{2}\right]+\left(\beta_{1}-\beta_{3}\right)^{2}$$

where $\beta_{1}+\beta_{3}=1-2 \beta_{4} ; 0 \leqslant \beta_{1}-\beta_{3} \leqslant \beta_{4}$.
When $\frac{1}{3} \leqslant \beta_{4} \leqslant \frac{1}{2}$, by equation (4) we have
$$S \leqslant 2\left[\beta_{4}^{2}+\left(1-\beta_{4}\right)^{2}\right]+\left(1-2 \beta_{4}\right)^{2} \leqslant \max \left\{\frac{11}{9}, 1\right\}=\frac{11}{9}$$

When $\frac{1}{4} \leqslant \beta_{4} \leqslant \frac{1}{3}$, by equation (4) we also have
$$S \leqslant 2\left[\beta_{4}^{2}+\left(1-\beta_{4}\right)^{2}\right]+\left(4 \beta_{4}-1\right)^{2} \leqslant \max \left\{\frac{11}{9}, \frac{10}{8}\right\}=\frac{10}{8}=\frac{5}{4}$$

Finally, combining equations (2), (5), and (6), we get $S \leqslant \frac{5}{4}$.
On the other hand, when $x_{1}=0, x_{2}=\frac{1}{4}, x_{3}=\frac{1}{2}, x_{4}^{-}=\frac{3}{4}$, we have $S=\frac{5}{4}$. Therefore, the minimum value of $a$ is $\frac{5}{4}$.