Let $n\ge5$ be an integer. Find the biggest integer $k$ such that there always exists a $n$-gon with exactly $k$ interior right angles. (Find $k$ in terms of $n$).
Problem
Source: Swiss Imo Selection 2006
Tags: geometry, algebra proposed, algebra
Xixas
26.05.2006 22:30
This problem originates from ISL 2003. Though it is still unsolved on the Mathlinks. See: http://www.mathlinks.ro/Forum/viewtopic.php?p=19086#p19086
BogG
26.05.2006 23:21
oke thanks
TheBernuli
17.01.2014 21:02
Sum of exterior angles in $n$-gon is always $360^{o}$, So $k=4$.
JBL
17.01.2014 21:35
4 is not achievable for all $n$. Also, the problem says nothing about convexity, so more than 4 is achievable for many $n$. (For example, consider the 12-gon in the shape of a Swiss cross, with 8 internal right angles.) (Perhaps a moderator could move this to a geometry forum, where it should have been placed originally 7+ years ago.)
TheBernuli
18.01.2014 03:41
Sorry for mistake. I thought that $n$-gon must be convex.