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

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-60345

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

A passenger left his things in an automatic storage room, and when he came to get his things, it turned out that he had forgotten the code. He only remembers that in the code there were the numbers 23 and 37. To open the room, you need to correctly type a five-digit number. What is the least number of codes you need to sort through in order to open the room for sure?

Problem #PRU-60363

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

11–12 2.0

A rectangular table is given, in each cell of which a real number is written, and in each row of the table the numbers are arranged in ascending order. Prove that if you arrange the numbers in each column of the table in ascending order, then in the rows of the resulting table, the numbers will still be in ascending order.

Problem #PRU-60410

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

Think of a way to finish constructing Pascal’s triangle upward.

Problem #PRU-60417

Problems Algebra and arithmetic Algebraic equations and systems of equations

11–13 2.0

Show that any natural number $n$ can be uniquely represented in the form $n = (\binom{x}{1}) + (\binom{y}{2}) + (\binom{z}{3})$ where $x, y, z$ are integers such that $0 \leq x < y < z$ , or $0 = x = y < z$ .

Problem #PRU-60437

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

11–12 2.0

Each side in the triangle $A B C$ is divided into 8 equal segments. How many different triangles exist with the vertices at the points of division (the points $A$ , $B$ , $C$ cannot be the vertices of triangles) in which neither side is parallel to either side of the triangle $A B C$ ?

Problem #PRU-60440

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

11–12 2.0

How many integers are there from 1 to 1,000,000, which are neither full squares, nor full cubes, nor numbers to the fourth power?

Problem #PRU-60449

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

Consider a chess board of size $n \times n$ . It is required to move a rook from the bottom left corner to the upper right corner. You can move only up and to the right, without going into the cells of the main diagonal and the one below it. (The rook is on the main diagonal only initially and in the final moment in time.) How many possible routes does the rook have?

Problem #PRU-60450

Problems Combinatorics

11–13 2.0

Tickets cost 50 cents, and $2 n$ buyers stand in line at a cash register. Half of them have one dollar, the rest – 50 cents. The cashier starts selling tickets without having any money. How many different orders of people can there be in the queue, such that the cashier can always give change?

Problem #PRU-60452

Problems Combinatorics

11–13 2.0

Prove that the Catalan numbers satisfy the recurrence relationship $C_{n} = C_{0} C_{n - 1} + C_{1} C_{n - 2} + \dots + C_{n - 1} C_{0}$ . The definition of the Catalan numbers $C_{n}$ is given in the handbook.