Question 5
The Fibonacci sequence $F_0,F_1,F_2,\ldots$ is defined by $F_0=0$, $F_1=1$, and $F_{n+1}=F_n+F_{n-1}$ for every $n\ge1$. Given $n\ge2$, determine the smallest possible size of a set $S$ of integers such that, for each $k=2,3,\ldots,n$, there exist $x,y\in S$ with $x-y=F_k$.