Let $I_n=\sum_{i=1}^n\sum_{j=1}^n\sum_{k=1}^n \min \left( \frac{1}{i}, \frac{1}{j}, \frac{1}{k} \right)$ and let $H_n=1+\frac{1}{2}+\ldots \frac{1}{n}$ Find $I_n-H_n$ in terms of $n$ (Paraphrased)
Source: Canada Junior MO 2024/2
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Let $I_n=\sum_{i=1}^n\sum_{j=1}^n\sum_{k=1}^n \min \left( \frac{1}{i}, \frac{1}{j}, \frac{1}{k} \right)$ and let $H_n=1+\frac{1}{2}+\ldots \frac{1}{n}$ Find $I_n-H_n$ in terms of $n$ (Paraphrased)