We are given 2001 balloons and a positive integer k. Each balloon has been blown up to a certain size (not necessarily the same for each balloon). In each step it is allowed to choose at most k balloons and equalize their sizes to their arithmetic mean. Determine the smallest value of k such that, whatever the initial sizes are, it is possible to make all the balloons have equal size after a finite number of steps. inductioncalculusintegrationcombinatorics unsolvedcombinatorics