On a ring road at regular intervals there are 25 posts, each with a policeman. The police are numbered in some order from 1 to 25. It is required that they cross the road so that there is a policeman on each post, but so that number 2 was clockwise behind number 1, number 3 was clockwise behind number 2, and so on. Prove that if you organised the transition so that the total distance travelled was the smallest, then one of the policemen will remain at his original post.
The numerical function \(f\) is such that for any \(x\) and \(y\) the equality \(f (x + y) = f (x) + f (y) + 80xy\) holds. Find \(f(1)\) if \(f(0.25) = 2\).
Solve the system of equations: \[\begin{aligned} \sin y - \sin x &= x-y; &&\text{and}\\ \sin y - \sin z &= z-y; && \text{and}\\ x-y+z &= \pi. \end{aligned}\]
Each day, from Monday to Friday, an old man went to the sea and threw in a net to catch fish. On each day the man caught no more fish than on the previous day. In total over the 5 days the man caught exactly 100 fish. What is the minimum total number of fish the man could have caught on Monday, Wednesday, and Friday.
Valentina added a number (not equal to 0) taken to the power of four and the same number to the power two and reported the result to Peter. Can Peter determine the unique number that Valentina chose?
In a row there are 20 different natural numbers. The product of every two of them standing next to one another is the square of a natural number. The first number is 42. Prove that at least one of the numbers is greater than 16,000.
The numbers \(1, 2, 3,\dots , 10\) are written around a circle in a particular order. Peter calculated the sum of each of the 10 possible groups of three adjacent numbers around the circle and wrote down the smallest value he had calculated. What is the largest possible value he could have written down?
In a group of six people, any five can sit down at a round table so that every two neighbours know each other.
Prove that the entire group can be seated at the round table so that every two neighbours will know each other.
Author: I.I. Bogdanov
Peter wants to write down all of the possible sequences of 100 natural numbers, in each of which there is at least one 3, and any two neighbouring terms differ by no more than 1. How many sequences will he have to write out?
A carpet has a square shape with side 275 cm. A moth has eaten 4 holes through it. Will it always be possible to cut a square section of side 1 m out of the carpet, so that the section does not contain any holes? Treat the holes as points.