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

In a room, there are three-legged stools and four-legged chairs. When people sat down on all of these seats, there were 39 legs $($human and stool/chair legs$)$ in the room. How many stools are there in the room?

Valerie wrote the number 1 on the board, and then several more numbers. As soon as Valerie writes the next number, Mike calculates the median of the already available set of numbers and writes it in his notebook. At some point, in Mike’s notebook, the numbers: 1; 2; 3; 2.5; 3; 2.5; 2; 2; 2; 2.5 are written.

a) What is the fourth number written on the board?

b) What is the eighth number written on the board?

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.

In a box of 2009 socks there are blue and red socks. Can there be some number of blue socks that the probability of pulling out two socks of the same colour at random is equal to 0.5?

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

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$)$.

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

In a room there are some chairs with 4 legs and some stools with 3 legs. When each chair and stool has one person sitting on it, then in the room there are a total of 39 legs. How many chairs and stools are there in the room?

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.

A pawn stands on one of the squares of an endless in both directions chequered strip of paper. It can be shifted by m squares to the right or by n squares to the left. For which m and n can it move to the next cell to the right?

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.

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