Problems

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.)

There are 20 students in a class, and each one is friends with at least 14 others. Can you prove that there are four students in this class who are all friends?

Each of the three axes has one rotating pin and a fixed arrow. The gears are connected in series. On the first gear there are 33 teeth, on the second – 10, on the third – 7. On each tooth of the first gear one symbol or letter of the following string of letters and symbols is written in the clockwise direction in the following order:

A B V C D E F G H I J K L M N O P Q R S T U W X Y Z ! ? \(>\) \(<\) $ £ €

On the teeth of the second and third gears in increasing order the numbers 0 to 9 and 0 to 6 are written respectively in a clockwise direction. When the arrow of the first axis points to a letter, the arrows of the other two axes point to numbers.

The letters and symbols of the message are encrypted in sequence. Encryption is performed by rotating the first gear anti-clockwise until the first possible letter or symbol that can be encrypted is landed on by the arrow. At this point, the numbers indicated by the second and third arrows are consistently written out. At the beginning of the encryption, the 1st wheel points to the letter A, and the arrows of the 2nd and 3rd wheels to the number 0.

Encrypt the Slavic name OLIMPIADA.

A message is encrypted using numbers where each number corresponds to a different letter of the alphabet. Decipher the following encoded text:

1317247191772413816720713813920257178

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).

Prove that there is no polyhedron that has exactly seven edges.

To transmit messages by telegraph, each letter of the Russian alphabet () ( and are counted as identical) is represented as a five-digit combination of zeros and ones corresponding to the binary number of the given letter in the alphabet (letter numbering starts from zero). For example, the letter is represented in the form 00000, letter -00001, letter -10111, letter -11111. Transmission of the five-digit combination is made via a cable containing five wires. Each bit is transmitted on a separate wire. When you receive a message, Cryptos has confused the wires, so instead of the transmitted word, a set of letters is received. Find the word you sent.