IMO 1993 - Shortlisted Problems Question 1

(BRA 1) Show that there exists a finite set $A\subset\mathbb R^2$ such that for every $X\in A$ there are points $Y_1,Y_2,\ldots,Y_{1993}$ in $A$ such that the distance between $X$ and $Y_i$ is equal to $1$, for every $i$.