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#### Boundedness, monotonicity , Quadratic inequaities and systems of inequalities

For which natural K does the number reach its maximum value?

#### Examples and counterexamples. Constructive proofs , Pigeonhole principle (other) , Theory of algorithms (other)

The function F is given on the whole real axis, and for each x the equality holds: F $(x + 1)$ F $(x)$ + F $(x + 1)$ + 1 = 0.
Prove that the function F can not be continuous.

#### 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?

#### Boundedness, monotonicity , Sequnces

$a_1$, $a_2$, $a_3$, … is an increasing sequence of natural numbers. It is known that $a_{a_k}$ = 3k for any k. Find a$)$ $a_{100}$; b$)$ $a_{1983}$.

#### 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.

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