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Are there such irrational numbers a and b so that a $>$ 1, b $>$ 1, and [$a^m$] is different from [$b^n$] for any natural numbers m and n?

n numbers are given as well as their product, p. The difference between p and each of these numbers is an odd number.

Prove that all n numbers are irrational.

Given an endless piece of chequered paper with a cell side equal to one. The distance between two cells is the length of the shortest path parallel to cell lines from one cell to the other $($it is considered the path of the centre of a rook$)$. What is the smallest number of colours to paint the board $($each cell is painted with one colour$)$, so that two cells, located at a distance of 6, are always painted with different colours?

Are the sum and product odd or even for:

a$)$ two even numbers?

b$)$ two odd numbers?

c$)$ an odd and an even number?

In the town of Ely, all families have separate houses. On one fine day, each family moved into a house that used to be occupied by another family. In this regard, it was decided to paint all houses in red, blue or green, and so that for each family the colour of the new and old houses did not match. Can this be done?

Two classes with the same number of students took a test. Having checked the test, the strict teacher Mr Jones said that he gave out 13 more twos than other marks $($where the marks range from 2 to 5 and 5 is the highest$)$. Was Mr Jones right?

On an island there are 1,234 residents, each of whom is either a knight $($who always tells the truth$)$ or a liar $($who always lies$)$. One day, all of the inhabitants of the island were broken up into pairs, and each one said: “He is a knight!” or “He is a liar!” about his partner. Could it eventually turn out to be that the number of “He is a knight!” and “He is a liar!” phrases is the same?

The numbers $1, 2, 3, …, 99$ are written onto 99 blank cards in order. The cards are then shuffled and then spread in a row face down. The numbers $1, 2, 3, .., 99$ are once more written onto in the blank side of the cards in order. For each card the numbers written on it are then added together. The 99 resulting summations are then multiplied together. Prove that the result will be an even number.

Are there functions p $($x$)$ and q $($x$)$ such that p $($x$)$ is an even function and p $($q $($x$)$$)$ is an odd function $($different from identically zero$)$?

A high rectangle of width 2 is open from above, and the G-shaped domino falls inside it in a random way $($see the figure$)$.

a) k G-shaped dominoes have fallen. Find the mathematical expectation of the height of the resulting polygon.

b) 7 G-shaped dominoes fell inside the rectangle. Find the probability that the resulting figure will have a height of 12.

An abstract artist took a wooden 5x5x5 cube and divided each face into unit squares. He painted each square in one of three colours – black, white, and red – so that there were no horizontally or vertically adjacent squares of the same colour. What is the smallest possible number of squares the artist could have painted black following this rule? Unit squares which share a side are considered adjacent both when the squares lie on the same face and when they lie on adjacent faces.

On a calculator, there are numbers from 0 to 9 and signs of two actions $($see the figure$)$. First, the display shows the number 0. You can press any key. The calculator performs the actions in the sequence of clicks. If the action sign is pressed several times, the calculator will only remember the last push. The Scattered Scientist pressed a lot of buttons in a random sequence. Find approximately the probability with which the outcome of the resulting chain of actions is an odd number?

On a calculator keypad, there are the numbers from 0 to 9 and signs of two actions $($see the figure$)$. First, the display shows the number 0. You can press any keys. The calculator performs the actions in the sequence of clicks. If the action sign is pressed several times, the calculator will only remember the last click.

a) The button with the multiplier sign breaks and does not work. The Scattered Scientist pressed several buttons in a random sequence. Which result of the resulting sequence of actions is more likely: an even number or an odd number?

b) Solve the previous problem if the multiplication symbol button is repaired.

16 teams took part in a handball tournament where a victory was worth 2 points, a draw – 1 point and a defeat – 0 points. All teams scored a different number of points, and the team that ranked seventh, scored 21 points. Prove that the winning team drew at least once.

a$)$ There are 21 coins on a table with the tails side facing upwards. In one operation, you are allowed to turn over any 20 coins. Is it possible to achieve the arrangement were all coins are facing with the heads side upwards in a few operations?

b$)$ The same question, if there are 20 coins, but you are allowed to turn over 19.

Could the difference of two integers multiplied by their product be equal to the number 1999?

Around a table sit boys and girls. Prove that the number of pairs of neighbours of different sexes is even.

Given a board (divided into squares) of the size: a) 10×12, b) 9×10, c) 9×11, consider the game with two players where: in one turn a player is allowed to cross out any row or any column if there is at least one square not crossed out. The loser is the one who cannot make a move. Is there a winning strategy for one of the players?

$2n$ diplomats sit around a round table. After a break the same $2n$ diplomats sit around the same table, but this time in a different order.

Prove that there will always be two diplomats with the same number of people sitting between them, both before and after the break.