The case of the diminishing circles...

Geometry Level 5

Let \(A_{0}\) be a unit circle centered at the origin. Draw another unit circle \(A_{1}\) centered at \((0,2)\), i.e., 'above' and tangent to \(A_{0}\).

Next, moving clockwise, draw a circle \(A_{2}\) of radius \(\frac{1}{2}\) that is externally tangent to both \(A_{0}\) and \(A_{1}\).

Next, continuing to move clockwise, draw a circle \(A_{3}\) of radius \(\frac{1}{3}\) that is externally tangent to \(A_{0}\) and \(A_{2}\).

Continue this process to create a series of circles \(A_{n}\) of radius \(\frac{1}{n}\) and externally tangent to \(A_{0}\) and \(A_{n-1}\), (and to \(A_{n+1}\), once it is drawn).

Let \(A_{k}\) be the last circle that can be drawn in this series before \(A_{k+1}\) partially overlaps \(A_{1}\). Find \(k\).

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