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#### 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.

#### Counting in two ways , Number tables and its properties

Is it possible to fill a $5 \times 5$ table with numbers so that the sum of the numbers in each row is positive and the sum of the numbers in each column is negative?

#### Counting in two ways , Number tables and its properties

In each square of a rectangular table of size $M \times K$, a number is written. The sum of the numbers in each row and in each column, is 1. Prove that M = K.

#### Counting in two ways , Partitions into pairs and groups bijections , Pigeonhole principle (other)

You are given 25 numbers. The sum of any 4 of these numbers is positive. Prove that the sum of all 25 numbers is also positive.

#### Counting in two ways , Partitions into pairs and groups bijections , Pigeonhole principle (other)

The total age of a group of 7 people is 332 years. Prove that it is possible to choose three members of this group so that the sum of their ages is no less than 142 years.

#### Algebraic inequalities (other) , Counting in two ways , Integer and fractional parts. Archimedean property

Prove that for any positive integer n the inequality is true.

#### Algebraic methods , Counting in two ways

Out of $7$ integer numbers, the sum of any $6$ is a multiple of $5$. Show that every one of these numbers is a multiple of $5$.

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