# 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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