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.
In a line 40 signs are written out: 20 crosses and 20 zeros. In one move, you can swap any two adjacent signs. What is the least number of moves in which it is guaranteed that you can ensure that some 20 consecutive signs are crosses?
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: \[\begin{aligned} \sin y - \sin x &= x-y; &&\text{and}\\ \sin y - \sin z &= z-y; && \text{and}\\ x-y+z &= \pi. \end{aligned}\]
To a certain number, we add the sum of its digits and the answer we get is 2014. Give an example of such a number.
Author: E.V. Bakaev
After a hockey match Anthony said that he scored 3 goals, and Ilya only one. Ilya said that he scored 4 goals, and Serge scored 5 goals. Serge said that he scored 6 goals, and Anthony only two. Could it be that the three of them scored 10 goals, if it is known that each of them once told the truth, and once lied?
Find all of the solutions of the puzzle: \(ARKA + RKA + KA + A = 2014\). (Different letters correspond to different numbers, and the same letters correspond to the same numbers.)
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.