Problem - 2471
Randomly colour all the points one a plane either black or white. Show that if any two points with a distance of $2$ units have the same colour, then all the points on this plane have the same colour.
If the conclusion does not hold, then there must exist two points of different colors with distance less than $4$. (If their distance is greater than $4$, it is always possible to advance one of of them towards the other at the step of $2$ until it meets the requirement.)
Let these two points be $A$ and $B$. Construct an isosceles triangle $ABC$ so that $AC=BC=2$. Then regardless of the color point $C$ has, it can not be the same as both $A$ and $B$. This is a contradiction.