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?
Once upon a time there were twenty spies. Each of them wrote an accusation against ten of his colleagues. Prove that at least ten pairs of spies have told on each other.
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?
To test a new program, a computer selects a random real number \(A\) from the interval \([1, 2]\) and makes the program solve the equation \(3x + A = 0\). Find the probability that the root of this equation is less than \(0.4\).
In a shopping centre, three machines sell coffee. During the day, the first machine can break down with a probability of 0.4 and the second with a probability of 0.3. Every evening, Mr Ivanov, the mechanic, comes and repairs all of the broken-down coffee machines. One day, Ivanov wrote, in his report, that the mathematical expectation of breakdowns during one week is 12. Prove that Mr Ivanov is exaggerating.
When one scientist comes up with an ingenious idea, he writes it down on a piece of paper, but then he realises that the idea is not brilliant, scrunches up this sheet of paper and throws it under the table, where there are two rubbish bins. The scientist misses the first bin with a probability \(p > 0.5\), and with the same probability he misses the second. In the morning, the scientist threw five crumpled brilliant ideas under the table. Find the probability that there was at least one of these ideas in each bin.
The television game “What? Where? When?” consists of a team of “experts” trying to solve 13 questions (or sectors), numbered from 1 to 13, that are thought up and sent in by the viewers of the programme. Envelopes with the questions are selected in turn in random order with the help of a spinning top with an arrow. If this sector has already come up previously, and the envelope is no longer there, then the next clockwise sector is played. If it is also empty, then the next one is played, etc., until there is a non-empty sector.
Before the break, the players played six sectors.
a) What is more likely: that sector number 1 has already been played or that sector number 8 has already been plated?
b) Find the probability that, before the break, six sectors with numbers from 1 to 6 were played consecutively.
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 \(2/3\)?
There were 50 white and black crows sitting on a birch, and the number of black crows was not less than the number of whites. On the oak, there too were white and black crows, and there were 50 of them in total. On the oak, the number of black crows was also not less than the number of white ones. It could be that there was the same number of black and white crows, or maybe even there was one black crow less than white crows. One random crow flew from the birch to the oak, and after a while another random crow (maybe the same one) flew from the oak to the birch. Which is more probable: that the number of white crows on the birch is the same as it was at first, or that it has changed?
At the Antarctic station, there are \(n\) polar explorers, all of different ages. With the probability \(p\) between each two polar explorers, friendly relations are established, regardless of other sympathies or antipathies. When the winter season ends and it’s time to go home, in each pair of friends the senior gives the younger friend some advice. Find the mathematical expectation of the number of those who did not receive any advice.