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Composite Figures

Overlap, inscribe, and circumscribe a collection of simple geometric shapes to make a complex, composite figure.

Composite Figures: Level 4 Challenges

         

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

Give your answer to 3 decimal places.


Inspiration.

In a triangle \(ABC\), a random line passes through its centroid (the intersection of the three medians), segmenting it into two regions. Find the minimum possible ratio of the area of the smaller region to the area of the larger region.

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 \(m+n\).

Let \(ABCD\) be a square and \(P\) be a point inside the square such that \(PB = 23\) and \(PD = 29\) . Find the area of \(\triangle APC\)

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