2024 IMO Problems/Problem 1

Determine all real numbers $\alpha$ such that, for every positive integer $n$ , the integer

$\lfloor \alpha \rfloor + \lfloor 2\alpha \rfloor + \dots +\lfloor n\alpha \rfloor$

is a multiple of $n$ . (Note that $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$ . For example, $\lfloor -\pi \rfloor = -4$ and $\lfloor 2 \rfloor = \lfloor 2.9 \rfloor = 2$ .)

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2024 IMO Problems/Problem 1