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#### Integer and fractional parts. Archimedean property , Pigeonhole principle (angles and lengths)

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.

#### Integer and fractional parts. Archimedean property , Pigeonhole principle (angles and lengths)

a$)$ Give an example of a positive number a such that {a} + {1 / a} = 1.
$\\$
b$)$ Can such an a be a rational number?

#### Integer and fractional parts. Archimedean property , Pigeonhole principle (angles and lengths)

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.

#### Integer and fractional parts. Archimedean property , Pigeonhole principle (angles and lengths)

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.

#### Integer and fractional parts. Archimedean property , Pigeonhole principle (angles and lengths)

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.

#### Integer and fractional parts. Archimedean property , Pigeonhole principle (angles and lengths)

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

#### Orthogonal projection , Pigeonhole principle (angles and lengths) , Polygonal chains inside a square , Triangle inequality (other)

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.

#### Circle, sector, segment, etc , Covers , Pigeonhole principle (angles and lengths) , Pigeonhole principle (finite number of poits, lines etc.)

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?

#### Measurement of segments and angles. Adjacent angles. , Pigeonhole principle (angles and lengths)

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 $26^{\circ}$ .

#### Integer and fractional parts. Archimedean property , Pigeonhole principle (angles and lengths)

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$)$.

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