We solve problems

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

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 Algorithm Theory Algorithm Theory

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-35708

Problems Calculus Real numbers Integer and fractional parts. Archimedean property Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

13–15 3.0

The positive irrational numbers $a$ and $b$ are such that $1/a + 1/b = 1$. Prove that among the numbers $\lfloor ma\rfloor , \lfloor nb\rfloor$ each natural number occurs exactly once.

Problem #PRU-35710

Problems Probability and statistics Probability theory Probability theory (other) Set theory and logic Algorithm Theory Algorithm Theory

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 Algorithm Theory Algorithm Theory

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-35719

Problems Methods Algebraic methods Partitions into pairs and groups; bijections Pigeonhole principle Pigeonhole principle (other)

12–14 3.0

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

Problem #PRU-35733

Problems Algebra Rational functions Identical transformations Set theory and logic Algorithm Theory Algorithm Theory

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-35734

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

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?

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-35753

Problems Combinatorics Integer lattices Integer lattices (other) Methods Pigeonhole principle Pigeonhole principle (other)

13–15 3.0

A convex polygon on a plane contains no fewer than $m^2+1$ points with whole number co-ordinates. Prove that within the polygon there are $m+1$ points with whole number co-ordinates that lie on a single straight line.