2019 IMO Shortlist C7

There are 60 empty boxes $B_{1}, \ldots, B_{60}$ in a row on a table and an unlimited supply of pebbles. Given a positive integer $n$, Alice and Bob play the following game. In the first round, Alice takes $n$ pebbles and distributes them into the 60 boxes as she wishes. Each subsequent round consists of two steps: (a) Bob chooses an integer $k$ with $1 \leq k \leq 59$ and splits the boxes into the two groups $B_{1}, \ldots, B_{k}$ and $B_{k+1}, \ldots, B_{60}$. (b) Alice picks one of these two groups, adds one pebble to each box in that group, and removes one pebble from each box in the other group. Bob wins if, at the end of any round, some box contains no pebbles. Find the smallest $n$ such that Alice can prevent Bob from winning. (Czech Republic)

桌子上一排有 60 个空盒子 $B_{1}, \ldots, B_{60}$，并且卵石无限供应。给定一个正整数 $n$，Alice 和 Bob 玩以下游戏。在第一轮中，爱丽丝取出 $n$ 鹅卵石，并按照自己的意愿将它们分配到 60 个盒子中。随后的每一轮都包含两个步骤：(a) Bob 选择一个整数 $k$，其中 $1 \leq k \leq 59$，并将盒子分为两组 $B_{1}, \ldots, B_{k}$ 和 $B_{k+1}, \ldots, B_{60}$。 (b) 爱丽丝选择这两组中的一组，向该组中的每个盒子添加一颗卵石，并从另一组中的每个盒子中移除一颗卵石。如果在任何一轮结束时，某个盒子里没有卵石，鲍勃就获胜。找到最小的$n$，这样爱丽丝就可以阻止鲍勃获胜。 (捷克共和国)