[ Extremal properties (miscellaneous).]

What is the largest number of points that can be placed on a segment of length 1 so that on any segment of length $d$, contained in this segment, there are no more than $1+1000 d^{2}$ points?
First, let's prove that it is impossible to place 33 points in such a way. Indeed, if 33 points are placed on a segment of length 1, then the distance between some two of them does not exceed $1 / 32$. The segment with endpoints at these points contains two points, but it should contain no more than $1 + 1000 / 32^{2}$ points, i.e., fewer than two points.

| Now let's prove that 32 points can be placed. Take 32 points that divide the segment into equal parts (the endpoints of the given segment are included in these 32 points). Then a segment of length $d$ contains either $[31d]$ or $[31d] + 1$ points. We need to prove that $[31 d] \leq 1000 d^{2}$. If $31 d < 1$, then $[31 d] = 0 < 1000 d^{2}$. If $31 d \geq 1$, then $[31 d] \leq 31 d \leq (31d)^{2} = 961 d^{2} < 1000 d^{2}$

Note $[x]$ - the integer part of the number $x$, i.e., the largest integer not exceeding $x$.