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A square is cut by 18 straight lines, 9 of which are parallel to one side of the square and the other 9 parallel to the other – perpendicular to the first 9 – dividing the square into 100 rectangles. It turns out that exactly 9 of these rectangles are squares. Prove that among these 9 squares there will be two that are identical.

Some real numbers $ a_1, a_2, a_3,…,a _{1996} $ are written in a row. Prove that it is possible to pick one or several adjacent numbers, so that their sum is less than 0.001 away from a whole number.

4 points $a, b, c, d$ lie on the segment $[0, 1]$ of the number line. Prove that there will be a point $x$, lying in the segment $[0, 1]$, that satisfies

$$\frac{1}{\left | x-a\right |}+\frac{1}{\left | x-b\right |}+\frac{1}{\left | x-c\right |}+\frac{1}{\left | x-d\right |} < 40 $$

We consider a sequence of words consisting of the letters “A” and “B”. The first word in the sequence is “A”, the k-th word is obtained from the $(k-1)$-th by the following operation: each “A” is replaced by “AAB” and each “B” by “A”. It is easy to see that each word is the beginning of the next, thus obtaining an infinite sequence of letters: AABAABAAABAABAAAB …

$\\$ a) Where in this sequence will the 1000th letter “A” be?

$\\$ b) Prove that this sequence is non-periodic.

a$)$ Give an example of a positive number a such that {a} + {1 / a} = 1.

$\\$

b$)$ Can such an a be a rational number?

f$(x)$ is an increasing function defined on the interval [0, 1]. It is known that the range of its values belongs to the interval [0, 1]. Prove that, for any natural N, the graph of the function can be covered by N rectangles whose sides are parallel to the coordinate axes so that the area of each is $1/N^2$. $($In a rectangle we include its interior points and the points of its boundary$)$.

Given an endless piece of chequered paper with a cell side equal to one. The distance between two cells is the length of the shortest path parallel to cell lines from one cell to the other $($it is considered the path of the centre of a rook$)$. What is the smallest number of colours to paint the board $($each cell is painted with one colour$)$, so that two cells, located at a distance of 6, are always painted with different colours?

At the cat show, 10 male cats and 19 female cats sit in a row where next to each female cat sits a fatter male cat. Prove that next to each male cat is a female cat, which is thinner than it.

The numbers a and b are such that the first equation of the system

$cos x = ax + b$

$sin x + a = 0$

has exactly two solutions. Prove that the system has at least one solution.

The numbers a and b are such that the first equation of the system

$sin x + a = bx$

$cos x = b$

has exactly two solutions. Prove that the system has at least one solution.

Aladdin visited all of the points on the equator, moving to the east, then to the west, and sometimes instantly moving to the diametrically opposite point on Earth. Prove that there was a period of time during which the difference in distances traversed by Aladdin to the east and to the west was not less than half the length of the equator.

Is there a line on the coordinate plane relative to which the graph of the function $y = 2^x$ is symmetric?

In a square with side length 1 there is a broken line, which does not self-intersect, whose length is no less than 200. Prove that there is a straight line parallel to one of the sides of the square that intersects the broken line at a point no less than 101 units along the line.

Two identical gears have 32 teeth. They were combined and 6 pairs of teeth were simultaneously removed. Prove that one gear can be rotated relative to the other so that in the gaps in one gear where teeth were removed there will be whole teeth of the second gear.

It is known that a camera located at $O$ cannot see the objects $A$ and $B$, where the angle $AOB$ is greater than $179^\circ$. 1000 such cameras are placed in a Cartesian plane. All of the cameras simultaneously take a picture. Prove that there will be a picture taken in which no more than 998 cameras are visible.

In a corridor of length 100m, 20 sections of red carpet are laid out. The combined length of the sections is 1000m. What is the largest number there can be of distinct stretches of the corridor that are not covered by carpet, given that the sections of carpet are all the same width as the corridor?

Four lamps need to be hung over a square ice-rink so that they fully illuminate it. What is the minimum height needed at which to hang the lamps if each lamp illuminates a circle of radius equal to the height at which it hangs?

You are given 7 straight lines on a plane, no two of which are parallel. Prove that there will be two lines such that the angle between them is less than $26^{\circ}$ .

Numbers 1, 2, 3, …, 101 are written out in a row in some order. Prove that one can cross out 90 of them so that the remaining 11 will be arranged in their magnitude $($either increasing or decreasing$)$.

Author: L.N. Vaserstein

For any natural numbers $a_1, a_2, …, a_m$, no two of which are equal to each other and none of which is divisible by the square of a natural number greater than one, and also for any integers and non-zero integers $b_1, b_2, …, b_m$ the sum is not zero. Prove this.