Geometry
# Composite Figures

What is the minimum side length of a square which can contain 5 non-overlapping unit circles?

Give your answer to 3 decimal places.

A star is inscribed inside a circle with radius \(r\) as shown in the picture above. The formula that puts the area of the star \(S\) in terms of the circle's radius \(r\) can be expressed as follows:

\[ \large S = \dfrac{a \sqrt{10a-b\sqrt a}}c r^2 \; , \]

where \(a,b\) and \(c\) are positive integers with \(a\) square-free and \(a,c\) coprime.

Find the value of \(a+b+c\).

Two congruent squares \(ABCD\) and \(PQRS\) are positioned such that they share a common area defined by \(\Delta PQB\). The ratio of the area of \(\Delta PQB\) to the area of polygon \(AQRSPCD\) is \( \dfrac{3}{22} \).

If the side length of both squares is \(s\), then the perimeter of polygon \(AQRSPCD\) is \(\dfrac{m}{n}s\), where \(m\) and \(n\) are coprime positive integers. Find \(m+n\).

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