Question 19
Given a sequence $S$ of $1001$ positive real numbers, not necessarily distinct, and given a set $A$ of distinct positive integers, the allowed operation is: for some $k \in A$, select $k$ numbers of $S$, compute the average of the $k$ numbers (arithmetic mean) and replace each of the $k$ selected numbers by that average.
If $A$ is a set such that for every $S$ one can achieve, by means of a sequence of allowed operations, that the numbers are all equal, determine the smallest possible value of the largest element of $A$.
If $A$ is a set such that for every $S$ one can achieve, by means of a sequence of allowed operations, that the numbers are all equal, determine the smallest possible value of the largest element of $A$.