Question 1
Determine all real numbers $ \alpha $ such that the number
$$ \left\lfloor\alpha\right\rfloor+\left\lfloor2\alpha\right\rfloor+\cdots+\left\lfloor n\alpha\right\rfloor $$
is a multiple of n for every positive integer n. (Here $\lfloor z\rfloor$ denotes the greatest integer less than or equal to z.)
$$ \left\lfloor\alpha\right\rfloor+\left\lfloor2\alpha\right\rfloor+\cdots+\left\lfloor n\alpha\right\rfloor $$
is a multiple of n for every positive integer n. (Here $\lfloor z\rfloor$ denotes the greatest integer less than or equal to z.)