Let $X = \{x_1, x_2, \dots, x_n\}$ be a set of $n$ positive numbers. If the arithmetic mean $A = \frac{1}{n} \sum_{i=1}^n x_i$ and the harmonic mean $H = \frac{n}{\sum_{i=1}^n \frac{1}{x_i}}$, which of the following statements are always true for distinct positive numbers?