# Fun with circles

Geometry Level 3

Let $$C$$ be a circle with centre $$P_0$$ and $$AB$$ be a diameter of $$C$$. Suppose $$P_1$$ is the midpoint of the line segment $$P_0B$$, $$P_2$$ is the midpoint of the line segment $$P_1B$$ and so on. Let $$C_1, C_2, C_3 , \ldots$$ be circles with diameters $$P_0P_1,P_1P_2,P_2P_3,\ldots$$ respectively. Suppose the circles $$C_1, C_2, C_3, \ldots$$ are all shaded. If the ratio of the area of the unshaded portion of $$C$$ to that of the original $$C$$ be expressed as $$M:N$$, then find $$M+N$$.

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