Problems

Using five nines, arithmetic operations and exponentiation, form the numbers from 1 to 13.

One day a strange notebook was found on the stairs. It contained one hundred statements:

“There is exactly one incorrect statement in this notebook”;

“There are exactly two incorrect statements in this notebook”;

“There are exactly three incorrect statements in this notebook”;

...

“There are exactly one hundred incorrect statements in this notebook.”

Are any of these statements true, and if so, which ones?

Jack the goldminer extracted 9 kg of golden sand. Will he be able to measure 2 kg of sand in three goes with the help of scales: a) with two weights of 200 g and 50 g; b) with one weight of 200 g?

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?

What word is encrypted: 22212221265121? Each letter is replaced by its number in the English alphabet.

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)?

In one move, it is permitted to either double a number or to erase its last digit. Is it possible to get the number 14 from the number 458 in a few moves?

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.

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?

An entire set of dominoes, except for 0-0, was laid out as shown in the figure. Different letters correspond to different numbers, the same – the same. The sum of the points in each line is 24. Try to restore the numbers.