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What is pi? If the area of a circle is 49π, what is its circumference? If a circle is inscribed in a square, what is the ratio of the areas of the circle and the square?

A quarter-circle with radius \(R\) is drawn. Inside it, two semicircles and a circle are drawn as shown in the above figure. If the area of the grey colored region is \[\frac { \pi { R }^{ 2 } }{ X },\] find the value of \(X.\)

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\(\Delta ABC\) is inscribed in a circle such that \(864 \angle A = \angle B = \angle C\). If \(B\) and \(C\) are adjacent vertices of a regular \(n\)-gon inscribed in the circle, find \(n\).

**Details and Assumptions**

- The diagram above is (extremely) not to scale.

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In a circle with center \(O\), a chord \(AB\) is drawn such that \(\angle AOB = 120^\circ\). Radius \(AO=10\). A circle is drawn in the major arc such that it's radius is maximum, with center E, it touches the larger circle at point X as shown in figure.

The area of the shaded region can be expressed as \(\dfrac{a\pi +b\sqrt{3}}{c}\) for integers \(a,b,c\) with \(\gcd(a,c)=1\) What is the value of \(a+b+c\)?

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You are a farmer, with a round fenced field of radius \(10\) meters. You tie your goat to the fence with a rope, and realize that the goat can eat exactly \(\frac{1}{2}\) of the field. How long is the rope?

**Note:** You can use a computational solver for the last step of the problem.

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