Question 1
Determine all real numbers $\alpha$ such that the number
$$
\lfloor\alpha\rfloor+\lfloor 2 \alpha\rfloor+\cdots+\lfloor n \alpha\rfloor
$$
is a multiple of $n$ for every positive integer $n$. (Here $\lfloor z\rfloor$ denotes the greatest integer less than or equal to $z$.) For example, $\lfloor-\pi\rfloor=-4$ and $\lfloor2\rfloor=\lfloor2.9\rfloor=2$.
$$
\lfloor\alpha\rfloor+\lfloor 2 \alpha\rfloor+\cdots+\lfloor n \alpha\rfloor
$$
is a multiple of $n$ for every positive integer $n$. (Here $\lfloor z\rfloor$ denotes the greatest integer less than or equal to $z$.) For example, $\lfloor-\pi\rfloor=-4$ and $\lfloor2\rfloor=\lfloor2.9\rfloor=2$.