IMC 2007 · Problem 4

Let $G$ be a finite group. For arbitrary sets $U, V, W \subset G$, denote by $N_{UVW}$ the number of triples $(x,y,z) \in U \times V \times W$ for which $xyz$ is the unity.

Suppose that $G$ is partitioned into three sets $A$, $B$ and $C$ (i.e. sets $A$, $B$, $C$ are pairwise disjoint and $G = A \cup B \cup C$). Prove that $N_{ABC} = N_{CBA}$.