Moderator Note: At the time that this question was posted, it was from an ongoing contest. The relevant deadline has now passed.
For all rational numbers, we will have a stick of variable length extending along $x=R$ and atop this stick will be a circular "stone" centered at the point where the stick ends. No two such stick-and-stone(consisting of the stick wielding the stone at its centre) constructs for distinct rational numbers can touch or cover any parts of each other(a stick cannot tangent a stone and a stone cannot tangent another stone). Can we construct a set of stick-and-stone figures for all rational numbers ranging from 0 to 1 non-inclusive abiding by these rules? Why or why not? Note again that the heights of these structures can vary and that the radius of the stone must be less than the height of the stick.