Is partitioning a multiset into two multisets with equal averages NP-complete?
One NP-hard variant of the PARTITION problem is as follows:


  INSTANCE: $2k$ positive integers $a_1,\ldots,a_{2k}$ with $\sum_{i=1}^{2k}a_i=2A$
  QUESTION: Does there exist an index set $I\subseteq\{1,\ldots,2k\}$ with $|I|=k$ so that $\sum_{i\in I}a_i=A$?


Take the integers $a_1,\ldots,a_{2k}$ together with two copies of the integer $100kA$. It is straightforward to see that these $2k+2$ numbers allow a partition into two sets with equal averages, if and only if the original instance of PARTITION has answer yes.