TopsOJ Summer 2025 - Individual Round - Problem 11

Let H_n denote the harmonic mean of the first n positive integers. If

\sum_{k=5}^{\infty} \frac{H_kH_{k-1}}{2k(1+2+\dots+(k-1))} = \frac{m}{n} for relatively prime positive integers m and n, compute m+n. Note: the harmonic mean of a_1, a_2, ... a_n is \frac{n}{\frac{1}{a_1}+\frac{1}{a_2}+\dots+\frac{1}{a_n}}.