6. Determine the minimum value of $\sum_{k=1}^{50} x_{k}$, where the summation is done over all possible positive numbers $x_{1}, \ldots, x_{50}$ satisfying $\sum_{k=1}^{50} \frac{1}{x_{k}}=1$.
6. Answer: 2500
By Cauchy-Schwarz inequality,
$$
\sum_{k=1}^{50} 1 \leq \sqrt{\sum_{k=1}^{50} x_{k} \sum_{k=1}^{50} \frac{1}{x_{k}}},
$$
and equality holds if and only if $x_{k}=50$ for $k=1, \ldots, 50$. Therefore the required value is 2500.