2020 IMO Shortlist C4

The Fibonacci numbers $F_{0}, F_{1}, F_{2}, \ldots$ are defined inductively by $F_{0}=0, F_{1}=1$, and $F_{n+1}=F_{n}+F_{n-1}$ for $n \geq 1$. Given an integer $n \geq 2$, determine the smallest size of a set $S$ of integers such that for every $k=2,3, \ldots, n$ there exist some $x, y \in S$ such that $x-y=F_{k}$. (Croatia)

斐波那契数 $F_{0}、F_{1}、F_{2}、\ldots$ 由 $F_{0}=0、F_{1}=1$ 和 $F_{n+1}=F_{n}+F_{n-1}$(对于 $n \geq 1$)归纳定义。给定一个整数 $n \geq 2$，确定整数集合 $S$ 的最小大小，使得对于每个 $k=2,3, \ldots, n$ 都存在一些 $x, y \in S$，使得 $x-y=F_{k}$。 (克罗地亚)