Problems

A cryptogram is given:

Restore the numerical values of the letters under which all of the equalities are valid, if different letters correspond to different digits. Arrange the letters in order of increasing numerical value and to find the required string of letters.

The rook stands on the square a1 of a chessboard. For a move, you can move it by any number of cells to the right or up. The one who puts the rook on the h8 square will win. Who wins with the right strategy?

Let \(x\) be a natural number. Among the statements:

\(2x\) is more than 70;

\(x\) is less than 100;

\(3x\) is greater than 25;

\(x\) is not less than 10;

\(x\) is greater than 5;

three are true and two are false. What is \(x\)?

It is known that any person has at most 400,000 hairs on their head. Given that the population of London is not less than 8 million, prove that there are 20 Londoners with the same number of hairs on their heads.

The key of the cipher, called the “lattice”, is a rectangular stencil of size 6 by 10 cells. In the stencil, 15 cells are cut out so that when applied to a rectangular sheet of paper of size 6 by 10, its cut-outs completely cover the entire area of the sheet in four possible ways. The letters of the string (without spaces) are successively entered into the cut-outs of the stencil (in rows, in each line from left to right) at each of its four possible positions. Find the original string of letters if, after encryption, the following text appeared in the sheet of paper

Hannah has 10 employees. Each month, Hannah raises the salary by 1 pound of exactly nine of her employees (of her choice). How can Hannah raise the salaries to make them equal? (Salaries are an integer number of pounds.)

Three friends decide, by a coin toss, who goes to get the juice. They have one coin. How do they arrange coin tosses so that all of them have equal chances to not have to go and get the juice?

There are 8 glasses of water on the table. You are allowed to take any two of the glasses and make them have equal volumes of water (by pouring some water from one glass into the other). Prove that, by using such operations, you can eventually get all the glasses to contain equal volumes of water.

26 numbers are chosen from the numbers 1, 2, 3, ..., 49, 50. Will there always be two numbers chosen whose difference is 1?

A broken calculator carries out only one operation “asterisk”: \(a*b = 1 - a/b\). Prove that using this calculator it is possible to carry out all four arithmetic operations (addition, subtraction, multiplication, division).