# Lopsidedness

Geometry Level 5

Consider three circles $$A$$, $$B$$ and $$C$$, each passing through the points $$(-1,0)$$ and $$(1,0)$$. Each circle is cut into three regions by the other two circles. Let the lopsidedness of a circle be the area of the circle's largest region divided by the area of the circle.

Let $$p>0$$. Now let $$A$$ pass through $$(1,2p)$$, $$B$$ pass through $$(0,1)$$, and $$C$$ pass through $$(-1,-2p)$$. If the value of $$p$$ minimising the lopsidedness of circle $$B$$ satisfies the equation

$(p^2+1) \arctan \left( \dfrac{1}{p} \right) = p+\dfrac{\pi}{K}$

find $$K$$.

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