Let $p,q$ be odd primes. Is it always possible to partition $\{1,\ldots,pq\}$ into $p$ disjoint subsets $A_1,\cdots,A_q$ such that for all $i,j$ we have $|A_i|=|A_j|$ and $\sum_{x\in A_i}x=\sum_{y\in A_j}y$?
For example we can do that for $p=3$ and $q=5$.
Partition into 3 subsets: $\{1,4,10,12,13\},\{2,5,8,11,14\},\{3,6,7,9,15\}$
Partition into 5 subsets: $\{1,8,15\},\{2,9,13\},\{3,10,11\},\{4,6,14\},\{5,7,12\}$