Solution:
Let $x_{1}, x_{2}, \ldots, x_{1000}$ be an arbitrary rearrangement of the numbers $1,2, \ldots, 1000$. Set

$$S_{1} = x_{1} + x_{2} + \cdots + x_{50}, \ldots, S_{20} = x_{951} + x_{952} + \cdots + x_{1000}$$
Since $S_{1} + \cdots + S_{20} = 500500$, we have $S_{i} \geq \frac{500500}{20} = 25025$ for at least one index $i$.

On the other hand, if a number $B$ has the required property then we have $B \leq 25025$. To see this consider the rearrangement
$$1000, 1, 999, 2, \ldots, 501, 500$$
and take arbitrarily fifty consecutive numbers in it. If the first number is greater than $500$, then the sum of these fifty numbers is $25025$, otherwise it is $25000$. Hence $A = 25025$.