If the sum of a finite number of positive real numbers is $1$ and each of them is less than $x$, then those real numbers can be partitioned into $50$ sets (some of which may be empty) such that the sum of the elements in a set is at most $x.$ Find the least possible value of $x.$
It is easy to verify that $\frac{1}{50} \le x$, so I think that the least possible value might be $\frac{1}{50}$, but I haven't proved that.