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.
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?
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 4 or 5, and any two neighbouring terms differ by no more than 2. 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.
A pack of 36 cards was placed in front of a psychic face down. He calls the suit of the top card, after which the card is opened, shown to him and put aside. After this, the psychic calls out the suit of the next card, etc. The task of the psychic is to guess the suit as many times as possible. However, the card backs are in fact asymmetrical, and the psychic can see in which of the two positions the top card lies. The deck is prepared by a bribed employee. The clerk knows the order of the cards in the deck, and although he cannot change it, he can prompt the psychic by having the card backs arranged in a way according to a specific arrangement. Can the psychic, with the help of such a clue, ensure the guessing of the suit of
a) more than half of the cards;
b) no less than 20 cards?
Every day, James bakes a square cake size \(3\times3\). Jack immediately cuts out for himself four square pieces of size \(1\times1\) with sides parallel to the sides of the cake (not necessarily along the \(3\times3\) grid lines). After that, Sarah cuts out from the rest of the cake a square piece with sides, also parallel to the sides of the cake. What is the largest piece of cake that Sarah can count on, regardless of Jack’s actions?
A unit square is divided into \(n\) triangles. Prove that one of the triangles can be used to completely cover a square with side length \(\frac{1}{n}\).