Find all of the natural numbers that, when divided by 7, have the same remainder and quotient.
In the gymnasium, all students know at least one of the ancient languages – Greek or Latin, some – both languages. 85% of all children know the Greek language and 75% know Latin. How many students know both languages?
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?
What word is encrypted: 22212221265121? Each letter is replaced by its number in the English alphabet.
Write the first 10 prime numbers in a line. How can you remove 6 digits to get the largest possible number?
Is it possible to arrange 44 marbles into 9 piles, so that the number of marbles in each pile is different?
Is it possible to cut a square into four parts so that each part touches each of the other three (ie has common parts of a border)?
Can the following equality be true: \[K \times O \times T = A \times B \times C \times D \times E \times F\] if you substitute the letters with the numbers from 1 to 9? Different letters correspond to different numbers.
A page of a calendar is partially covered by the previous torn sheet (see the figure). The vertices A and B of the upper sheet lie on the sides of the bottom sheet. The fourth vertex of the lower leaf is not visible – it is covered by the top sheet. The upper and lower pages, of course, are identical in size to each other. Which part of the lower page is greater, that which is covered or that which is not?
Twenty-eight dominoes can be laid out in various ways in the form of a rectangle of \(8 \times 7\) cells. In Fig. 1–4 four variants of the arrangement of the figures in the rectangles are shown. Can you arrange the dominoes in the same arrangements as each of these options?