A cubic polynomial \(f (x)\) is given. Let’s find a group of three different numbers \((a, b, c)\) such that \(f (a)= b\), \(f (b) = c\) and \(f (c) = a\). It is known that there were eight such groups \([a_i, b_i, c_i]\), \(i = 1, 2, \dots , 8\), which contains 24 different numbers. Prove that among eight numbers of the form \(a_i + b_i + c_i\) at least three are different.
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.
Ten tennis players came to the competitions, 4 of them were from Russia. According to the rules for the first round, the tennis players are broken into pairs randomly. Find the probability that in the first round, all Russian tennis players will play only with other Russian tennis players.
In the triangle \(ABC\), the angle \(A\) is equal to \(40^{\circ}\). The triangle is randomly thrown onto a table. Find the probability that the vertex \(A\) lies east of the other two vertices.
At the power plant, rectangles that are 2 m long and 1 m wide are produced. The length of the objects is measured by the worker Howard, and the width, irrespective of Howard, is measured by the worker Rachel. The average error is zero for both, but Howard allows a standard measurement error (standard deviation of length) of 3 mm, and Rachel allows a standard error of 2 mm.
a) Find the mathematical expectation of the area of the resulting rectangle.
b) Find the standard deviation of the area of the resulting rectangle in centimetres squared.
At a factory known to us, we cut out metal disks with a diameter of 1 m. It is known that a disk with a diameter of exactly 1 m weighs exactly 100 kg. During manufacturing, a measurement error occurs, and therefore the standard deviation of the radius is 10 mm. Engineer Gavin believes that a stack of 100 disks on average will weigh 10,000 kg. By how much is the engineer Gavin wrong?
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.
At a conference there were 18 scientists, of which exactly 10 know the eye-popping news. During the break (coffee break), all scientists are broken up into random pairs, and in each pair, anyone who knows the news, tells this news to another if he did not already know it.
a) Find the probability that after the coffee break, the number of scientists who know the news will be 13.
b) Find the probability that after the coffee break the number of scientists who know the news will be 14.
c) Denote by the letter \(X\) the number of scientists who know the eye-popping news after the coffee break. Find the mathematical expectation of \(X\).