We solve problems

Problem #PRU-35676

Problems Algebra and arithmetic Rational functions Inequalities. Method of Intervals. Set theory and logic Mathematical logic Mathematical logic (other)

12–14 2.0

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 $x$?

Problem #PRU-35685

Problems Set theory and logic Mathematical logic Mathematical logic (other)

11–12 2.0

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

Problem #PRU-35703

Problems Set theory and logic Theory of algotithms Theory of algorithms (other)

12–13 2.0

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.)

Problem #PRU-35710

Problems Probability and statistics Probability theory Probability theory (other) Set theory and logic Theory of algotithms Theory of algorithms (other)

12–13 3.0

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?

Problem #PRU-35714

Problems Methods Mathematical induction Mathematical induction (other) Set theory and logic Theory of algotithms Theory of algorithms (other)

12–14 3.0

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.

Problem #PRU-35733

Problems Algebra and arithmetic Rational functions Identical transformations Set theory and logic Theory of algotithms Theory of algorithms (other)

12–14 3.0

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).

Problem #PRU-35745

Problems Set theory and logic Mathematical logic Mathematical logic (other)

11–12 2.0

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.

Problem #PRU-35761

Problems Set theory and logic Mathematical logic Mathematical logic (other)

11–12 2.0

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

1317247191772413816720713813920257178

Problem #PRU-35786

Problems Algebra and arithmetic Number systems Binary number system Algebraic geometry Cryptography Set theory and logic Theory of algotithms Theory of algorithms (other)

13–15 2.0

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.

Problem #PRU-35797

Problems Set theory and logic Theory of algotithms Theory of algorithms (other)

12–13 2.0

In one urn there are two white balls, in another two black ones, in the third – one white and one black. On each urn there was a sign indicating its contents: WW, BB, WB. Someone rehung the signs so that now each sign indicating the contents of the urn is incorrect. It is possible to remove a ball from any urn without looking into it. What is the minimum number of removals required to determine the composition of all three urns?