We are given 111 different natural numbers that do not exceed 500. Could it be that for each of these numbers, its last digit coincides with the last digit of the sum of all of the remaining numbers?
The number \(x\) is such a number that exactly one of the four numbers \(a = x - \sqrt{2}\), \(b = x-1/x\), \(c = x + 1/x\), \(d = x^2 + 2\sqrt{2}\) is not an integer. Find all such \(x\).
Peter marks several cells on a \(5 \times 5\) board. His friend, Richard, will win if he can cover all of these cells with non-overlapping corners of three squares, that do not overlap with the border of the square (you can only place the corners on the squares). What is the smallest number of cells that Peter should mark so that Richard cannot win?
In the isosceles triangle \(ABC\), the angle \(B\) is equal to \(30^{\circ}\), and \(AB = BC = 6\). The height \(CD\) of the triangle \(ABC\) and the height \(DE\) of the triangle \(BDC\) are drawn. Find the length \(BE\).
Three players are playing knockout table tennis – that is, the player who loses a game swaps places with the player who did not take part in that game and the winner stays on. In total Andrew played 10 games, Ben played 15, and Charlotte played 17. Which player lost the second game played?
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.
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.