Question 109
Find the integers $n$ for which the equality
$$
\sum_{i=1}^{n} \sum_{j=1}^{n}\left\lfloor\frac{i \times j}{n+1}\right\rfloor=\frac{n^{2}(n-1)}{4}
$$
holds.
Recall that, for every real number $x$, the notation $\lfloor x\rfloor$ denotes the integer part of $x$, that is, the largest integer less than or equal to $x$.
$$
\sum_{i=1}^{n} \sum_{j=1}^{n}\left\lfloor\frac{i \times j}{n+1}\right\rfloor=\frac{n^{2}(n-1)}{4}
$$
holds.
Recall that, for every real number $x$, the notation $\lfloor x\rfloor$ denotes the integer part of $x$, that is, the largest integer less than or equal to $x$.