BALKAN 2020 Problem 3

Let $k$ be a positive integer. Determine the least integer $n \geqslant k+1$ for which the game below can be played indefinitely:

Consider $n$ boxes, labelled $b_{1}, b_{2}, \ldots, b_{n}$. For each index $i$, box $b_{i}$ contains initially exactly $i$ coins. At each step, the following three substeps are performed in order: (1) Choose $k+1$ boxes; (2) Of these $k+1$ boxes, choose $k$ and remove at least half of the coins from each, and add to the remaining box, if labelled $b_{i}$, a number of $i$ coins. (3) If one of the boxes is left empty, the game ends; otherwise, go to the next step.