IMO Shortlist 1983 problem 25
Dodao/la:
arhiva2. travnja 2012. Prove that every partition of
![3](/media/m/b/8/2/b82f544df38f2ea97fa029fc3f9644e0.png)
-dimensional space into three disjoint subsets has the following property: One of these subsets contains all possible distances; i.e., for every
![a \in \mathbb R^+](/media/m/2/7/3/273769f13bcbdeb0171e5a7d8c081836.png)
, there are points
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
and
![N](/media/m/f/1/9/f19700f291b1f2255b011c11d686a4cd.png)
inside that subset such that distance between
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
and
![N](/media/m/f/1/9/f19700f291b1f2255b011c11d686a4cd.png)
is exactly
%V0
Prove that every partition of $3$-dimensional space into three disjoint subsets has the following property: One of these subsets contains all possible distances; i.e., for every $a \in \mathbb R^+$, there are points $M$ and $N$ inside that subset such that distance between $M$ and $N$ is exactly $a.$
Izvor: Međunarodna matematička olimpijada, shortlist 1983