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.
Fred chose 2017 (not necessarily different) natural numbers \(a_1, a_2, \dots , a_{2017}\) and plays by himself in the following game. Initially, he has an unlimited supply of stones and 2017 large empty boxes. In one move Fred adds a1 stones to any box (at his choice), in any of the remaining boxes (of his choice) – \(a_2\) stones, ..., finally, in the remaining box – \(a_{2017}\) stones. His purpose is to ensure that eventually all the boxes have an equal number of stones. Could he have chosen the numbers so that the goal could be achieved in 43 moves, but is impossible for a smaller non-zero number of moves?
Gary drew an empty table of \(50 \times 50\) and wrote on top of each column and to the left of each row a number. It turned out that all 100 written numbers are different, and 50 of them are rational, and the remaining 50 are irrational. Then, in each cell of the table, he wrote down a product of numbers written at the top of its column and to the left of the row (the “multiplication table”). What is the largest number of products in this table which could be rational numbers?
In one box, there are two pies with mushrooms, in another box there are two with cherries and in the third one, there is one with mushrooms and one with cherries. The pies look and weigh the same, so it’s not known what is in each one. The grandson needs to take one pie to school. The grandmother wants to give him a pie with cherries, but she is confused herself and can only determine the filling by breaking the pie, but the grandson does not want a broken pie, he wants a whole one.
a) Show that the grandmother can act so that the probability of giving the grandson a whole pie with cherries will be equal to \(2/3\).
b) Is there a strategy in which the probability of giving the grandson a whole pie with cherries is higher than \(2/3\)?
The triangle \(C_1C_2O\) is given. Within it the bisector \(C_2C_3\) is drawn, then in the triangle \(C_2C_3O\) – bisector \(C_3C_4\) and so on. Prove that the sequence of angles \(\gamma_n = C_{n + 1}C_nO\) tends to a limit, and find this limit if \(C_1OC_2 = \alpha\).