Let $\mathcal P$ be the set of all points in $\mathbb R^n$ with rational coordinates. For the points $A,B \in \mathcal l{P}$, one can move from $A$ to $B$ if the distance $AB$ is $1$. Prove that every point in $\mathcal l{ P}$ can be reached from any other point in $\mathcal{P}$ by a finite sequence of moves if and only if $n \geq 5$.
Problem
Source: Iran Third Round MO 1997, Exam 3, P6
Tags: analytic geometry, combinatorics proposed, combinatorics
19.10.2005 12:21
This one appeared many times here, and I solved it somewhere. Search... Pierre.
19.10.2005 12:22
Here it is : http://www.mathlinks.ro/Forum/viewtopic.php?highlight=lagrange&t=2635 Pierre.
13.11.2024 13:43
How to solve it?
13.11.2024 14:14
xyz123456 wrote: How to solve it? See the url shared by @pbornsztein. You can repair almost all the old, non-working urls just by bringing them to the form https://artofproblemsolving.com/community/c[x]h[y] where [x] is the id of the corresponding forum and [y] is the id of the topic. Specifically for our situation the url of @pbornsztein becomes https://artofproblemsolving.com/community/c6h2635.
13.11.2024 17:03
kiyoras_2001 wrote: xyz123456 wrote: How to solve it? See the url shared by @pbornsztein. You can repair almost all the old, non-working urls just by bringing them to the form https://artofproblemsolving.com/community/c[x]h[y] where [x] is the id of the corresponding forum and [y] is the id of the topic. Specifically for our situation the url of @pbornsztein becomes https://artofproblemsolving.com/community/c6h2635. I see why y=2635 here, but how do you know that x=6?
13.11.2024 17:10
Ianis wrote: I see why y=2635 here, but how do you know that x=6? When viewing a forum you can see in the url its id as c[x]. For High School Olympiads, for example, x=6. Or for College Math x=7. Sometimes in the old urls the forum id is also specified, though in this case it was not. I just guessed that it would have to be located in HSO. Usually this is the case when the forum id is omitted in the old urls.
13.11.2024 17:57
I see, thank you!