Regular Polygons

Regular Polygons: Level 4 Challenges


Inside a regular pentagon ABCDEABCDE, construct 5 more regular pentagons of side length 12AB\dfrac{1}{2}AB. The part of the overlapping of these pentagons yields another regular pentagon of side length MNMN.

Let r=ABMNr=\dfrac{AB}{MN}. Find 1000r\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 nthn^{\text{th}} figure in the above sequence is constructed by the following procedure:

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

Let AnA_n be the total blue area of the nthn^{\text{th}} figure in the sequence.

Compute limnAn\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 ABCDEFGHABCDEFGH has squares ACEGACEG and BDFHBDFH inscribed in it. These squares form a smaller octagon as shown.

Let the the area of octagon ABCDEFGHABCDEFGH be ALA_L and the area of the smaller octagon be ASA_S. Then for some integers aa and bb, where bb is square-free, ASAL=ab.\large \dfrac{A_S}{A_L}=a-\sqrt{b}. Find a+ba+b.


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