IMO Shortlist 2004 problem N7
Kvaliteta:
Avg: 0,0Težina:
Avg: 9,0 Let
be an odd prime and
a positive integer. In the coordinate plane, eight distinct points with integer coordinates lie on a circle with diameter of length
. Prove that there exists a triangle with vertices at three of the given points such that the squares of its side lengths are integers divisible by
.
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
![p^{n}](/media/m/5/1/5/515aee9afad0b11eb7d8865b17742184.png)
![p^{n+1}](/media/m/2/0/d/20db5cae796a04243920815fe147084f.png)
Izvor: Međunarodna matematička olimpijada, shortlist 2004