Pick arbitrarily one point A out of the given 25 and consider a circle B(A,1) centered at A and radius 1. If all the remaining points lie inside B(A,1), there is nothing to prove - B(A,1) is the circle we've been looking for.

Else, there is a point B that lies outside B(A,1). Let B(B,1) be a unit circle centered at B. By definition, the distance between A and B exceeds 1. For any point C, by the given condition, either the distance to A or the distance to B is less then 1. In other words, each of the remaining points belongs to either B(A,1) or B(B,1). By the Pigeonhole Principle, at least 13 lie in the same circle.