Question 175
Let $n$ be an even natural number. The numbers $1,2, \ldots, n^{2}$ are partitioned into two sets $A$ and $B$ of equal size, in such a way that each of the $n^{2}$ numbers belongs to exactly one of the two sets. Let $S_{A}$ and $S_{B}$ be the sum of all the elements in $A$ and $B$ respectively. Determine all $n$ for which there exists a partition such that
$$
\frac{S_{A}}{S_{B}}=\frac{39}{64}
$$
$$
\frac{S_{A}}{S_{B}}=\frac{39}{64}
$$