Oh, my ... They did not even try to disguise the question. I am totally opposed to asking such folklore questions in a competition, since some competitors might simply have come across this, no matter the level of scholar education in that country.

$\chi \ge 4$: Assume $\chi \le 3$. Considering equilateral triangles tessellation, we get any 2 points which are $\sqrt{3}$ apart has same color. So we can make a triangle of side lengths $1,\sqrt{3},\sqrt{3}$, hence contratiction. $\chi \le 7$: Consider tessellation with regular hexagons, which have diameter 0.999. So you can paint each hexagons satisfying the condition: For instance, continuing this pattern. 1 3 5 4 2 4 1 2 6 3 2 6 7 3 5 4 3 5 4 1 5 1 2 6