I was attempting to solve the following question:

In a circle you have a $27$ sided regular polygon and a $297$ sided polygon $($all vertices are on the circle$).$ How many common points do they share?

I thought that $n\choose r$ would be helpful but I don't have any idea about how to use it.

  • $\begingroup$ It is 22, obviously. $\endgroup$
    – HyJu
    Dec 24 '16 at 16:22
  • 2
    $\begingroup$ How @HyJu ????? $\endgroup$ Dec 24 '16 at 16:23
  • $\begingroup$ @THELONEWOLF. Clear. $\endgroup$
    – HyJu
    Dec 24 '16 at 16:25
  • $\begingroup$ Ah, I made a mistake. It is 54=2*27, not 22=2*11. $\endgroup$
    – HyJu
    Dec 24 '16 at 16:26
  • 1
    $\begingroup$ @HyJu You haven't shown that this maximum is attained. I also find your response to THE LONE WOLF to be very rude. You claimed that your incorrect answer was "obvious" and "clear," without providing any justification. $\endgroup$ Dec 24 '16 at 16:42

HINT: Both polygons consist of straight line segments (the edges), and both are convex. Hence each edge of the one polygon intersects at most two edges of the other polygon. (Prove this!)

  • $\begingroup$ So we always have the number $=2n$ where $m \geq n$ are the number of vertices of the two polygons. $\endgroup$
    – HyJu
    Dec 24 '16 at 17:03
  • $\begingroup$ @servaes, what is the role of circle ??? $\endgroup$ Dec 24 '16 at 17:04
  • $\begingroup$ @THELONEWOLF. The polygons inscribed in a circle must be convex. $\endgroup$
    – HyJu
    Dec 24 '16 at 17:05
  • $\begingroup$ The role of the circle is to make both polygons intersect eachother (a lot). Otherwise you could shrink one to fit inside the other entirely, or translate one to get as few intersections as you like. And indeed they force the polygons to be convex, though this also follows from regularity, otherwise this becomes a mess. $\endgroup$
    – Servaes
    Dec 24 '16 at 17:08
  • $\begingroup$ @HyJu, As far as I think, the purpose of the website is to create a friendly environment and learn math,. But you have vandalised both the purposes. I know that under the know circumstances,, my answer can be considered wrong (That is why I have temporarily deleted it) but I still believe that there is huge possibility that OP was asking for vertices. $\endgroup$ Dec 24 '16 at 17:19

I observed that a line of 27-gon have 11 sides of 297-gon in minor segment if we consider 1point of both polygons coincide.

Also when 1point coincide than each point of 27-gon coincide by simmatry

So only 27 points are common

If I consider no point coincide then 1line of 27-gon intersect with 2 sides of 297- gon so there are 54 points common.


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