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# Regular Polygons

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

# Regular Polygons: Level 4 Challenges

Inside a regular pentagon $$ABCDE$$, construct 5 more regular pentagons of side length $$\dfrac{1}{2}AB$$. The part of the overlapping of these pentagons yields another regular pentagon of side length $$MN$$.

Let $$r=\dfrac{AB}{MN}$$. Find $$\displaystyle \left \lfloor 1000r \right \rfloor$$.

Above shows a 18-sided regular polygon. How many obtuse triangles are there formed by 3 vertices?

###### Try part 1.

The $$n^{\text{th}}$$ figure in the above sequence is constructed by the following procedure:

1. Draw a blue disc of radius $$\displaystyle\sqrt{\frac{2016}{\pi}}$$
2. Remove a regular $$n$$-gon area from the (smallest) disc
3. Inscribe a blue disc inside the empty $$n$$-gon space
4. Repeat steps 2-4

Let $$A_n$$ be the total blue area of the $$n^{\text{th}}$$ figure in the sequence.

Compute $$\displaystyle\lim_{n\to\infty}A_n$$.

It is not easy to draw a regular decagon without tools.

On a piece of writing paper (with equally spaced lines), I am trying to draw a regular decagon, as shown above. I started by drawing two sides so that their vertical extent is precisely 1 unit of the paper (black lines).

Now I want to draw the next side (red line), and I wonder how far it will extend vertically. To 3 decimal places, what is the distance marked with a question mark?

A regular octagon $$ABCDEFGH$$ has squares $$ACEG$$ and $$BDFH$$ inscribed in it. These squares form a smaller octagon as shown.

Let the the area of octagon $$ABCDEFGH$$ be $$A_L$$ and the area of the smaller octagon be $$A_S$$. Then for some integers $$a$$ and $$b$$, where $$b$$ is square-free, $\large \dfrac{A_S}{A_L}=a-\sqrt{b}.$ Find $$a+b$$.

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