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A numerical sequence is defined by the following conditions:

\[a_1 = 1, \,\,\, a_{n+1} = a_n + ⌊\sqrt{a_n}⌋\]

$\\$

Prove that among the terms of this sequence there are an infinite number of complete squares.

Write the first 10 prime numbers in a line. How can you remove 6 digits to get the largest possible number?

Find all of the natural numbers that, when divided by 7, have the same remainder and quotient.

One three-digit number consists of different digits that are in ascending order, and in its name all words begin with the same letter. The other three-digit number, on the contrary, consists of identical digits, but in its name all words begin with different letters. What are these numbers?

In March 2015 a teacher ran 11 sessions of a maths club. Prove that if no sessions were run on Saturdays or Sundays there must have been three days in a row where a session of the club did not take place. The 1st March 2015 was a Sunday.

Is it possible to place the numbers $1, 2,…12$ around a circle so that the difference between any two adjacent numbers is 3, 4, or 5?

The point O is randomly chosen on piece of square paper. Then the square is folded in such a way that each vertex is overlaid on the point O. The figure shows one of the possible folding schemes. Find the mathematical expectation of the number of sides of the polygon that appears.

In the first term of the year Daniel received five grades in mathematics with each of them being on a scale of 1 to 5, and the most common grade among them was a 5 . In this case it turned out that the median of all his grades was 4, and the arithmetic mean was 3.8. What grades could Daniel have?

Prove that the 13th day of the month is more likely to occur on a Friday than on other days of the week. It is assumed that we live in the Gregorian style calendar.

The order of books on a shelf is called * wrong * if no three adjacent books are arranged in order of height (either increasing or decreasing). How many wrong orders is it possible to construct from $n$ books of different heights, if:

a) $n = 4$;

b) $n = 5$?

How many different four-digit numbers, divisible by 4, can be made up of the digits 1, 2, 3 and 4,

a) if each number can occur only once?

b) if each number can occur several times?

The KUB is a cube. Prove that the ball, CIR, is not a cube. $($KUB and CIR are three-digit numbers, where different letters denote different numbers$)$.

One day, Claudia, Sofia and Freia noticed that they brought the same toy cars to kindergarten. Claudia has a car with a trailer, a small car and a green car without a trailer. Sofia has a car without a trailer and a small green one with a trailer, and Freia has a big car and a small blue car with a trailer. What kind of car $($in terms of colour, size and availability of a trailer$)$ did all of the girls bring to the kindergarten? Explain the answer.

Peter has some coins in his pocket. If Peter pulls 3 coins from his pocket, without looking, there will always be a £1 coin among them. If Peter pulls 4 coins from his pocket, without looking, there will always be a £2 coin among them. Peter pulls 5 coins from his pocket. Identify these coins.

We are given a table of size $n \times n$. $n-1$ of the cells in the table contain the number $1$. The remainder contain the number $0$. We are allowed to carry out the following operation on the table:

1. Pick a cell.

2. Subtract 1 from the number in that cell.

3. Add 1 to every other cell in the same row or column as the chosen cell.

Is it possible, using only this operation, to create a table in which all the cells contain the same number?

A square napkin was folded in half, the resulting rectangle was then folded in half again $($see the figure$)$. The resulting square was then cut with scissors $($in a straight line$)$. Could the napkin have been broken up a$)$ into 2 parts? b$)$ into 3 parts? c$)$ into 4 parts? d$)$ into 5 parts? If yes – illustrate such a cut, if not – write the word “no”.

Find the smallest four-digit number CEEM for which there exists a solution to the rebus MN + PORG = CEEM. $($The same letters correspond to the same numbers, different – different$)$.

In any group of 10 children, out of a total of 60 pupils, there will be three who are in the same class. Will it always be the case that amongst the 60 pupils there will be:

1) 15 classmates?

2) 16 classmates?