The teacher on probability theory leaned back in his chair and looked at the screen. The list of those who signed up is ready. The total number of people turned out to be \(n\). Only they are not in alphabetical order, but in a random order in which they came to the class.
“We need to sort them alphabetically,” the teacher thought, “I’ll go down in order from the top down, and if necessary I’ll rearrange the student’s name up in a suitable place. Each name should be rearranged no more than once”.
Prove that the mathematical expectation of the number of surnames that you do not have to rearrange is \(1 + 1/2 + 1/3 + \dots + 1/n\).
Are there functions \(p (x)\) and \(q (x)\) such that \(p (x)\) is an even function and \(p (q (x))\) is an odd function (different from identically zero)?
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?
The function \(f (x)\) is defined for all real numbers, and for any \(x\) the equalities \(f (x + 2) = f (2 - x)\) and \(f (x + 7) = f (7 - x)\) are satisfied. Prove that \(f (x)\) is a periodic function.
A tennis tournament takes place in a sports club. The rules of this tournament are as follows. The loser of the tennis match is eliminated (there are no draws in tennis). The pair of players for the next match is determined by a coin toss. The first match is judged by an external judge, and every other match must be judged by a member of the club who did not participate in the match and did not judge earlier. Could it be that there is no one to judge the next match?
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 \(0.4\).