Two Children Play With Three Coins

Geometry Level 5

Two children collaborate on a problem for amusement. They place three coins, all of which are unit circles, inside a circular playing field of radius \(R\).

They take turns making a move, with the following provisions

1) One coin is moved in each turn, not necessarily (but may be) in a straight path, sliding it from where it is to a new position inside the circular playing field without disturbing the other two coins,

2) such that the center of the moved coin slides past the line connecting the centers of the other two coins (i.e., sliding between the other two coins) exactly once,

3) and to a new position not necessarily having to (but may) touch any of the other two coins or the border of the circular playing field.

4) The same coin may not be moved in two consecutive turns.

5) The two children deliberately play in such a way to make possible taking turns indefinitely, i.e., not being forced to stop playing because it has become impossible to move a coin and have its center pass through the line connecting centers of the other two coins without disturbing them.

6) The two children deliberately at the start place the three coins in the circular playing field in such a way that such indefinitely extended play is made possible.

What is the infimum radius \(R\) of the circular playing field for this to be possible?

Submit your answer as \(\left\lfloor 1000 R \right\rfloor \).

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