Question

IF YOU WILL ANSWER THIS QUESTION I WILL MAKE YOU IN BRAINLIST. 1. There are infinite black and white dots on a plane. Prove that the distance between one black dot and one white dot is one unit.

Answer 1

Answer:

Assuming every point of the plane is either white or black, here is a quick "constructive" way to find two points of opposite colour at distance 1. Since there are both white and black points, the infimum r of the distances between white and black points is well defined. If r>0 then there exist a black-white pair at distance d with r≤d<2r, and the midpoint between them is at distance d/2<r of either, so it cannot be black or white by the choice of r, a contradiction. So r=0, and there exists a black-white pair at distance d<1 of each other. The circles of radius 1 centered at these points intersect, and pairing the two centers with the two intersection points one gets at least one black-white pair at distance 1 (in fact one gets two pairs).

Answer 2

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There are infinite black and white dots on a plane. ... In either case we can see that plane is colored with black or white color only. So we can say that for each white dot there is exist a black dot at unit distance because if not than we end up coloring the full plane with white color.

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