Find all positive integers $m, n$, such that the inequality
$$
[(m+n) \alpha]+[(m+n) \beta] \geqslant[m \alpha]+[m \beta]+[n(\alpha+\beta)]
$$

holds for any real numbers $\alpha, \beta$.
If $m=n$, then let $n \alpha=x, n \beta=y$, by the previous problem, (1) holds.
If $n>m$, take $\alpha=\beta \in \left[\frac{1}{2 n}, \frac{1}{m+n}\right)$, then the left side of (1) $=0$, the right side $\geqslant[n(\alpha+$ $\beta)] \geqslant 1$.

If $m>n$, take $\alpha=\frac{1}{m}, \beta=\frac{m-1}{m}$, then the right side of (1) $=m+n$, the left side $=1+(m-1)+$ $\left[\frac{n(m-1)}{m}\right]<m+n$.
Therefore, the answer to this problem is $m=n$.