Consider a finite set $\mathbb{A} = \{ \frac12, \frac13, \frac14... \frac1n \} $ and I want to find all of the subsets of $\mathbb{A}, \mathbb{B}$, such that $$\sum_{x_i \in \mathbb{B}} x_i = 1 $$
A trivial solution is $\frac12 + \frac13 + \frac16$. If $n$ is a number big like $50$, there are $2^{50}$ possible subsets of A. Not even a computer would be able to brute force all of the possible subsets. Is there a more efficient approach or an algorithm to find these sets faster? What if A didn't contain the sequence $\frac1x$ but something like $\frac1{x^2}$ or $\frac1{\ln{x}}$ or some other function? Is there an approach that can be generalized for $\frac{1}{f(x)}$?