IMO Shortlist 1986 problem 16
Dodao/la:
arhiva2. travnja 2012. Let
![A,B](/media/m/7/1/7/7174f8a9f33236ee137c01b144237389.png)
be adjacent vertices of a regular
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
-gon (
![n\ge5](/media/m/0/2/d/02d675fa201b7accc9c95490e649b5ca.png)
) with center
![O](/media/m/9/6/0/9601b72f603fa5d15addab9937462949.png)
. A triangle
![XYZ](/media/m/1/3/d/13dab5022dd1d33f3d299852f2f54cfb.png)
, which is congruent to and initially coincides with
![OAB](/media/m/4/a/c/4ac8783af608ce16ab6fe8ecef768cd3.png)
, moves in the plane in such a way that
![Y](/media/m/3/b/c/3bc24c5af9ce86a9a691643555fc3fd6.png)
and
![Z](/media/m/7/9/4/794ff2bd637e30ea27e50e57eecd0b76.png)
each trace out the whole boundary of the polygon, with
![X](/media/m/9/2/8/92802f174fc4967315c2d8002c426164.png)
remaining inside the polygon. Find the locus of
![X](/media/m/9/2/8/92802f174fc4967315c2d8002c426164.png)
.
%V0
Let $A,B$ be adjacent vertices of a regular $n$-gon ($n\ge5$) with center $O$. A triangle $XYZ$, which is congruent to and initially coincides with $OAB$, moves in the plane in such a way that $Y$ and $Z$ each trace out the whole boundary of the polygon, with $X$ remaining inside the polygon. Find the locus of $X$.
Izvor: Međunarodna matematička olimpijada, shortlist 1986