We solve problems

Problem #PRU-32787

Problems Algebra and arithmetic Number systems Decimal number system Methods Algebraic methods Partitions into pairs and groups; bijections Pigeonhole principle Pigeonhole principle (other)

12–13 3.0

Is it possible to find 57 different two digit numbers, such that no sum of any two of them was equal to 100?

Problem #PRU-34976

Problems Probability and statistics Probability theory Discrete distribution Methods Algebraic methods Partitions into pairs and groups; bijections

12–16 4.0

Two people toss a coin: one tosses it 10 times, the other – 11 times. What is the probability that the second person’s coin showed heads more times than the first?

Problem #PRU-35149

Problems Geometry Plane geometry Geometrical inequalities Area inequalities Methods Algebraic methods Partitions into pairs and groups; bijections Pigeonhole principle Pigeonhole principle (area and volume)

13–15 3.0

In a square which has sides of length 1 there are 100 figures, the total area of which sums to more than 99. Prove that in the square there is a point which belongs to all of these figures.

Problem #PRU-35476

Problems Calculus Functions of one variable. Continuity Methods Algebraic methods Iterations

15–16 3.0

Let \(f (x)\) be a polynomial about which it is known that the equation \(f (x) = x\) has no roots. Prove that then the equation \(f (f (x)) = x\) does not have any roots.

Problem #PRU-35643

Problems Methods Algebraic methods Partitions into pairs and groups; bijections Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Geometry Plane geometry Polygons Regular polygons

13–14 4.0

2001 vertices of a regular 5000-gon are painted. Prove that there are three coloured vertices lying on the vertices of an isosceles triangle.

Problem #PRU-35667

Problems Algebra and arithmetic Algebraic geometry Cryptography Methods Algebraic methods Partitions into pairs and groups; bijections

13–14 2.0

The key of the cipher, called the “swivelling grid”, is a stencil made from a square sheet of chequered paper of size \(n \times n\) (where \(n\) is even). Some of the cells are cut out. One side of the stencil is marked. When this stencil is placed onto a blank sheet of paper in four possible ways (marked side up, right, down or left), its cut-outs completely cover the entire area of the square, where each cell is found under the cut-out exactly once. The letters of the message, that have length \(n^2\), are successively written into the cut-outs of the stencil, where the sheet of paper is placed on a blank sheet of paper with the marked side up. After filling in all of the cut-outs of the stencil with the letters of the message, the stencil is placed in the next position, etc. After removing the stencil from the sheet of paper, there is an encrypted message.

Find the number of different keys for an arbitrary even number \(n\).

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-35791

Problems Algebra and arithmetic Word problems Solving problems by inequalities. Going through the cases. Methods Extremal principle Algebraic methods Processes and operations

13–14 3.0

Sam and Lena have several chocolates, each weighing not more than 100 grams. No matter how they share these chocolates, one of them will have a total weight of chocolate that does not exceed 100 grams. What is the maximum total weight of all of the chocolates?