IZhO 2006 Problem 4

In a pile you have 100 stones. A partition of the pile in $k$ piles is good if:

1. the small piles have different numbers of stones;

2. for any partition of one of the small piles in 2 smaller piles, among the $k + 1$ piles you get 2 with the same number of stones (any pile has at least 1 stone).

Find the maximum and minimal values of $k$ for which this is possible.