CONO-SUR 2009 Problem 5

Given a succession $C$ of $1001$ positive real numbers (not necessarily distinct), and given a set $K$ of distinct positive integers, the permitted operation is: select a number $k\in{K}$ , then select $k$ numbers in $C$ , calculate the arithmetic mean of those $k$ numbers, and replace each of those $k$ selected numbers with the mean.

If $K$ is a set such that for each $C$ we can reach, by a sequence of permitted operations, a state where all the numbers are equal, determine the smallest possible value of the maximum element of $K$ .