IMC 2023 · Problem 8

Let $T$ be a tree with $n$ vertices; that is, a connected simple graph on $n$ vertices that contains no cycle. For every pair $u, v$ of vertices, let $d(u, v)$ denote the distance between $u$ and $v$, that is, the number of edges in the shortest path in $T$ that connects $u$ with $v$.

Consider the sums

$$W(T) = \sum_{\substack{\{u,v\} \subseteq V(T) \\ u \ne v}} d(u, v) \quad \text{and} \quad H(T) = \sum_{\substack{\{u,v\} \subseteq V(T) \\ u \ne v}} \frac{1}{d(u, v)}.$$

Prove that

$$W(T) \cdot H(T) \ge \frac{(n-1)^3(n+2)}{4}.$$