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 \(K\).

This is a slightly rephrased version of a question part that appeared in an Oxford Mathematics Admissions Test.

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