Kubarev A.M.

From a chessboard of 8x8, 10 cells have been cut out. It is known that among the cut-out cells, there are both black and white cells. What is the maximum number of two-cell rectangles that can be guaranteed to be cut out from this board after this?

#
Each two-cell rectangle contains a black and a white cell, so if 9 white cells are cut out, it is impossible to cut out more than $32-n$ dominoes.

![](https://cdn.mathpix.com/cropped/2024_05_06_7c6a12be356bd13b62cag-12.jpg?height=469&width=471&top_left_y=1606&top_left_x=16)

Fig. 1

![](https://cdn.mathpix.com/cropped/2024_05_06_7c6a12be356bd13b62cag-12.jpg?height=192&width=240&top_left_y=1602&top_left_x=709)

$\downarrow$

![](https://cdn.mathpix.com/cropped/2024_05_06_7c6a12be356bd13b62cag-12.jpg?height=188&width=249&top_left_y=1887&top_left_x=710)

Fig. 2

Let's cut the board as shown in Fig. 1. The cells cut out of the board during the cut will "ruin" no more than 10 rectangles. Therefore, we already have at least 22 intact rectangles. We will show how to increase the number of intact rectangles by 1. Consider the closed chain of cells depicted in Fig. 1 (we move along the chain from cell a2 upwards). Since both white and black cells are cut out, there must be a cut white cell followed by a cut black cell in this chain. If these cells are adjacent, they "ruin" only one rectangle, meaning that with such a cut, there will be no fewer than 23 intact rectangles. Otherwise, if there are other cells between them, we will divide the board between them so that the new rectangle starts immediately after the cut white cell (see Fig. 2). Then the number of intact rectangles will increase by 1. Therefore, there will again be no fewer than 23 intact rectangles.

## Answer

23.

Author: Blynkov A.D.

Given an isosceles trapezoid, the sum of whose lateral sides is equal to the larger base. Prove that the acute angle between the diagonals is no more than .

## Solution

First solution. Consider a trapezoid with bases and, where $A D=$ BARreulvnik $A B$

$$
\sin \angle A D B=\frac{A H}{A D} \leq \frac{A B}{A D}=\frac{1}{2}
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_7c6a12be356bd13b62cag-13.jpg?height=54&width=1812&top_left_y=602&top_left_x=17)
the angle between the diagonals, $\angle$ AOBahiygOmenyshiy angle is no more than , as the adjacent angle.

Second solution. The trapezoid is inscribed in a circle. Its lateral side is half the length of the base and, therefore, is no longer than the radius of the circle. Therefore, the lateral sides subtend arcs no more than . And the angle between the diagonals is equal to half the sum of these arcs.