There are 23 students in a class. During the year, each student of this class celebrated their birthday once, which was attended by some (at least one, but not all) of their classmates. Could it happen that every two pupils of this class met each other the same number of times at such celebrations? (It is believed that at every party every two guests met, and also the birthday person met all the guests.)
A mix of boys and girls are standing in a circle. There are 20 children in total. It is known that each boys’ neighbour in the clockwise direction is a child wearing a blue T-shirt, and that each girls’ neighbour in the anticlockwise direction is a child wearing a red T-shirt. Is it possible to uniquely determine how many boys there are in the circle?
In a \(10 \times 10\) square, all of the cells of the upper left \(5 \times 5\) square are painted black and the rest of the cells are painted white. What is the largest number of polygons that can be cut from this square (on the boundaries of the cells) so that in every polygon there would be three times as many white cells than black cells? (Polygons do not have to be equal in shape or size.)
Author: A. Glazyrin
In the coordinate space, all planes with the equations \(x \pm y \pm z = n\) (for all integers \(n\)) were carried out. They divided the space into tetrahedra and octahedra. Suppose that the point \((x_0, y_0, z_0)\) with rational coordinates does not lie in any plane. Prove that there is a positive integer \(k\) such that the point \((kx_0, ky_0, kz_0)\) lies strictly inside some octahedron from the partition.
For the anniversary of the London Mathematical Olympiad, the mint coined three commemorative coins. One coin turned out correctly, the second coin on both sides had two heads, and the third had tails on both sides. The director of the mint, without looking, chose one of these three coins and tossed it at random. She got heads. What is the probability that the second side of this coin also has heads?
In a convex hexagon, independently of each other, two random diagonals are chosen. Find the probability that these diagonals intersect inside the hexagon (inside – that is, not at the vertex).
The shooter shoots at 3 targets until he shoots everything. The probability of a hit with one shot is \(p\).
a) Find the probability that he needs exactly 5 shots.
b) Find the mathematical expectation of the number of shots.
One hundred cubs found berries in the forest: the youngest managed to grab 1 berry, the next youngest cub – 2 berries, the next – 4 berries, and so on, until the oldest who got \(2^{99}\) berries. The fox suggested that they share the berries “fairly.” She can approach two cubs and distribute their berries evenly between them, and if this leaves an extra berry, then the fox eats it. With such actions, she continues, until all the cubs have an equal number of berries. What is the largest number of berries that the fox can eat?
Hannah Montana wants to leave the round room which has six doors, five of which are locked. In one attempt she can check any three doors, and if one of them is not locked, then she will go through it. After each attempt her friend Michelle locks the door, which was opened, and unlocks one of the neighbouring doors. Hannah does not know which one exactly. How should she act in order to leave the room?
There are 30 students in a class: excellent students, mediocre students and slackers. Excellent students answer all questions correctly, slackers are always wrong, and the mediocre students answer questions alternating one by one correctly and incorrectly. All the students were asked three questions: “Are you an excellent pupil?”, “Are you a mediocre student?”, “Are you a slacker?”. 19 students answered “Yes” to the first question, to the second 12 students answered yes, to the third 9 students answered yes. How many mediocre students study in this class?