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#### 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?

#### Counting in two ways , Exponential functions and logarithms (other) , Integer and fractional parts. Archimedean property

Prove that for every natural number n $>$ 1 the equality: [$n^{1 / 2}] + [n^{1/ 3}] + … + [n^{1 / n}] = [log_{2}n] + [log_{3}n] + … + [log_{n}n]$ is satisfied.

#### 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.

#### Equations of higher order (other) , Integer and fractional parts. Archimedean property

During the chess tournament, several players played an odd number of games. Prove that the number of such players is even.

#### 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?

#### Divisibility of a number. General properties , Examples and counterexamples. Constructive proofs , Integer and fractional parts. Archimedean property

Does there exist a number h such that for any natural number n the number [$h \times 1969^n$] is not divisible by [$h \times 1969^{n-1}$]?

#### Equations of higher order (other) , Integer and fractional parts. Archimedean property

How can you connect 50 cities with the least number of airlines so that from every city you can get to any other one by making no more than two transfers?

#### Equations of higher order (other) , Integer and fractional parts. Archimedean property

On an 8 × 8 chessboard the largest possible number of bishops is placed so that no two bishops threaten each other. Prove that the number of all such constellations is an exact square.

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