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.)
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.)
In a convex polygon, which has an odd number of vertices equal to \(2n + 1\), two independently of each other random diagonals are chosen. Find the probability that these diagonals intersect inside the polygon.
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?
10 children were each given a bowl with 100 pieces of pasta. However, these children did not want to eat and instead started to play. One of the children started to place one piece of pasta into every other child’s bowl. What is the least amount of transfers needed so that everyone has a different number of pieces of pasta in their bowl?
100 children were each given a bowl with 100 pieces of pasta. However, these children did not want to eat and instead started to play. One of the children started to place one piece of her pasta into other children’s bowls (to whomever she wants). What is the least amount of transfers needed so that everyone has a different number of pieces of pasta in their bowl?
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?
100 switched on and 100 switched off lights are randomly arranged in two boxes. Each flashlight has a button, the button of which turns off an illuminated flashlight and switches on a turned off flashlight. Your eyes are closed and you can not see if the flashlight is on. But you can move the flashlights from a box to another box and press the buttons on them. Think of a way to ensure that the burning flashlights in the boxes are equally split.
Replace the letters with numbers (all digits must be different) so that the correct equality is obtained: \(A/ B/ C + D/ E/ F + G/ H/ I = 1\).