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#### Equations in integer numbers , Integer and fractional parts. Archimedean property

PFind the number of solutions in natural numbers of the equation [x / 10] = [x / 11] + 1.

#### Equations in integer numbers , Statistics

In the set -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, replace one number with two other integers so that the set variance and its mean remain unchanged.

#### Equations in integer numbers , Sorting in increasing (decreasing) order

Prove that the equation $\frac {x}{y}$ + $\frac {y}{z}$ + $\frac {z}{x}$ = 1 is unsolvable using positive integers.

#### Equations in integer numbers , Extremal principle (other) , Geometric interpretations in algebra , Pigeonhole principle (other) , The greatest common divisor (GCD) and the least common multiplier (LCM). Mutually prime numbers

Let a, b, c be integers; where a and b are not equal to zero.
Prove that the equation ax + by = c has integer solutions if and only if c is divisible by d = GCD $($a, b$)$.

#### Equations in integer numbers , Integer and fractional parts. Archimedean property

How many solutions in natural numbers does the equation [x / 10] = [x / 11] + 1 have?

#### Equations in integer numbers , Pigeonhole principle (other)

There are 100 notes of two types: a and b pounds, and a $\neq$ b $($mod 101$)$.
Prove that you can select several bills so that the amount received (in pounds) is divisible by 101.

#### Equations in integer numbers , Rational and irrational numbers

The real numbers x and y are such that for any distinct prime odd p and q the number $x^p$ + $y^q$ is rational. Prove that x and y are rational numbers.

#### Equations in integer numbers , Joke problem

Is it possible to find natural numbers $x$, $y$ and $z$ which satisfy the equation $28x+30y+31z=365$?

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