US-TST 2016 Problem 5

Let $n \geq 4$ be an integer. Find all functions $W:\{1, \ldots, n\}^{2} \rightarrow \mathbb{R}$ such that for every partition $[n]=A \cup B \cup C$ into disjoint sets, $$\sum_{a \in A} \sum_{b \in B} \sum_{c \in C} W(a, b) W(b, c)=|A||B||C| .$$