10.53 Can 1965 points be placed in a square with side length 1, such that any rectangle inside the square, with sides parallel to those of the square and area $\frac{1}{200}$, contains at least one of these points?
[Solution] In the square $0 \leqslant x, y \leqslant 1$, 200 points are uniformly placed on the median line $y=\frac{1}{2}$:
\[ C_{0}=\left\{\left.\left(\frac{k}{201}, \frac{1}{2}\right) \right\rvert\, k=1,2, \cdots, 200\right\}. \]
Then, 100 points are placed on each of the lines $y=\frac{1}{4}$ and $y=\frac{3}{4}$, making a total of 200 points:
\[ C_{1}=\left\{\left.\left(\frac{k}{101}, \frac{1}{4}\right) \right\rvert\, k=1,2, \cdots, 100\right\} \cup \left\{\left.\left(\frac{k}{101}, \frac{3}{4}\right) \right\rvert\, k=1,2, \cdots, 100\right\}. \]
This process continues, and for $m=2,3, \cdots, 7$, points are uniformly placed on each line $y=(2 l-1) 2^{-m-1}, 1 \leqslant l \leqslant 2^{m}$:
\[ m_{m}=\left\{\left.\left(\frac{k}{m_{m}+1}, \frac{2 l-1}{2^{m+1}}\right) \right\rvert\, k=1,2, \cdots, m_{m}\right\}, \]
where $m_{m}=\left[200 \cdot 2^{-m}\right]$. When $m=7$, one point is placed on each of the 128 corresponding line segments. Thus, the total number of points placed is:
\[ \begin{aligned}
\sum_{m=0}^{7} 2^{m} m_{m}= & 200+2 \times 100+4 \times 50+8 \times 25+16 \times 12+32 \times 6 \\
& +64 \times 3+128=1704 .
\end{aligned} \]
This process can be visualized in the figure above.
It is easy to verify that among the 1704 points placed, no rectangle with an area of $\frac{1}{200}$ can be squeezed in. In fact, if the rectangle intersects the median line $y=\frac{1}{2}$, its base is no more than $\frac{1}{201}$, and its area is no more than $1 \times \frac{1}{201}$; if it does not intersect the median line but intersects the line $y=\frac{1}{4}$ or $y=\frac{3}{4}$, its height is no more than $\frac{1}{2}$, and its base is no more than $\frac{1}{101}$, and so on. Finally, if the rectangle does not intersect any line of the form $y=\frac{n}{256}, n=1,2, \cdots, 255$, its height is no more than $\frac{1}{256}$.