IMO 2024 Problem 1

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

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

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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$\alpha$ qualifies $\Leftrightarrow$ it is an even integer. It suffices to restrict $\alpha$ to $[0,2)$ because of periodicity. No odd integer $\alpha$ works because $3\alpha$ is not even.

For $\alpha\in[0,1)$, by setting $n=2$ we get $\alpha\in[0, 0.5)$. When $n=3$, it is further restricted to $\alpha\in[0, 1/3)$. We keep going on, where each time we find ourselves further requiring $\alpha\in[0,1/n)$. Therefore we have $\alpha=0$.

For $\alpha\in(1,2)$, by setting $n=2$ we get $\alpha\in[1.5,2)$. When $n=3$, it is further restricted to $\alpha\in[5/3,2)$. We keep going on, where each time we find ourselves further requiring $\alpha\in[2-1/n,2)$ because $(n-1)^2+2n-1=n^2$. Therefore $\alpha$ does not exist and we conclude that $\alpha$ must be an even integer.