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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.

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    An example would be useful to clarify what you mean. What is the stick and stone representation for $2/3$?2012-03-05
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    Somehow I think the Farey sequence of rationals will be useful here. Or else you want to construct an enumeration of the rationals and assign a value to each rational such that for any two rationals, the sum of the values assigned to them is less than the distance between them.2012-03-05
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    A line of some length along x=2/3, topped with a circle centered at the endpoint of that line that is not on the x-axis, or a radius less than the length of the line.2012-03-05
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    There is a proof that states that no interval of any size $\epsilon$ is without a rational number. Perhaps that might help?2012-03-05
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    Are these lines vertical or horizontal?2012-03-05
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    As I said they run along x=R meaning that they are vertical.2012-03-05

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