ILL 1974 Problem 51

There are $n$ points on a flat piece of paper, any two of them at a distance of at least $2$ from each other. An inattentive pupil spills ink on a part of the paper such that the total area of the damaged part equals $\frac 32$ . Prove that there exist two vectors of equal length less than $1$ and with their sum having a given direction, such that after a translation by either of these two vectors no points of the given set remain in the damaged area.