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With equal angles and equal side lengths, what more could you want from a polygon?

The diagram above shows a regular hexagon \({ H }_{3 }\) with area \(H\) which has six right triangles inscribed in it. Let the area of the shaded region be \(S\), then what is the ratio \(H:S\)?

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Above figures shows a unit square \(ABCD\).

If the area of the octagon \(EFGHIJKL\) (in blue) can be expressed as \(\dfrac{1}{a}\) , find \(a\).

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As shown in the image above, a pentagon, hexagon and decagon are inscribed in three congruent circles, and their endpoints are connected to form a triangle. If the radii of each of the circles is \(1\) and the area of the triangle formed by the three polygons can be written as \(\frac{\sqrt{a}-b}{c}\), where \(a\), \(b\) and \(c\) are coprime integers, what is \(a+b+c\)?

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We can make "*pencilogons*" by aligning multiple identical pencils end of tip to start of tip together without leaving any gaps, as shown above, so that the enclosed area forms a regular polygon (the example above left is an 8-*pencilogon*).

Hazri wants to make an \(n\)-*pencilogon* using \(n\) identical pencils with pencil tips of angle \(7^\circ.\) After he aligned \(n-18\) pencils, he found out the gap between the two ends is too small to fit in another pencil.

So, in order to complete the *pencilogon*, he has to sharpen all the \(n\) pencils so that the angle of all the pencil tips becomes \((7-m)^\circ\).

Find the value of \(m+n\).

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