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Regular Polygons

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

Level 4

         

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?


To enlarge the image, click here.
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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