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In a 10-storey house, 1 person lives on the first floor, 2 on the second floor, 3 on the third, 4 on the fourth, … 10 on the tenth. On which floor does the elevator stop most often?

The angle at the top of a crane is 20$^{\circ}$. How will the magnitude of this angle change when looking at the crane with binoculars which triple the size of everything?

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?

Find the coefficient of x for the polynomial $($x – a$)$ $($x – b$)$ $($x – c$)$ … $($x – z$)$.

Andrew drives his car at a speed of 60 km/h. He wants to travel every kilometre 1 minute faster. By how much should he increase his speed?

A message is encrypted by replacing the letters of the source text with pairs of digits according to some table $($known only to the sender and receiver$)$ in which different letters of the alphabet correspond to different pairs of digits. The cryptographer was given the task to restore the encrypted text. In which case will it be easier for him to perform the task: if it is known that the first word of the second line is a “thermometer” or that the first word of the third line is “smother”? Justify your answer. $($It is assumed that the cryptographic table is not known$)$.

There are two purses and one coin. Inside the first purse is one coin, and inside the second purse is one coin. How can this be?

One and a half diggers dig for a half hour and end up having dug half a pit. How many pits will two diggers dig in two hours?

With the help of scissors, cut a hole in a notebook through which an elephant could climb!

Is it possible to arrange 6 long round pencils so that each of them touches all the other ones?

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

Henry did not manage to get into the elevator on the first floor of the building and decided to go up the stairs. It takes 2 minutes to rise to the third floor. How long does it take to rise to the ninth floor?

Homework. Cut a hole in an exercise book of a size so that you yourself can climb through it.

We call a natural number “amazing” if it has the form $a^b$ + $b^a$ $($where a and b are natural numbers$)$. For example, the number 57 is amazing, since 57 =$2^5$ + $5^2$. Is the number 2006 amazing?

Gerard says: the day before yesterday I was 10 years old, and next year I will turn 13. Can this be?

In two purses lie two coins, and one purse has twice as many coins as the other. How can this be?

Can the equality $K \times O \times T$ = $U \times W \times E \times N \times H \times Y$ be true if the numbers from 1 to 9 are substituted for letters instead of the letters? Different letters correspond to different numbers.

In a bookcase, there are four volumes of the collected works of Astrid Lindgren, with each volume containing 200 pages. A worm who lives on this bookshelf has gnawed its way from the first page of the first volume to the last page of the fourth volume. Through how many pages has the worm gnawed its way through?