There are \(n\) random vectors of the form \((y_1, y_2, y_3)\), where exactly one random coordinate is equal to 1, and the others are equal to 0. They are summed up. A random vector a with coordinates \((Y_1, Y_2, Y_3)\) is obtained.
a) Find the mathematical expectation of a random variable \(a^2\).
b) Prove that \(|a|\geq \frac{1}{3}\).
On one island, one tribe has a custom – during the ritual dance, the leader throws up three thin straight rods of the same length, connected in the likeness of the letter capital \(\pi\), \(\Pi\). The adjacent rods are connected by a short thread and therefore freely rotate relative to each other. The bars fall on the sand, forming a random figure. If it turns out that there is self-intersection (the first and third bars cross), then the tribe in the coming year are waiting for crop failures and all sorts of trouble. If there is no self-intersection, then the year will be successful – satisfactory and happy. Find the probability that in 2019, the rods will predict luck.
An incredible legend says that one day Stirling was considering the numbers of Stirling of the second kind. During his thoughtfulness, he threw 10 regular dice on the table. After the next throw, he suddenly noticed that in the dropped combination of points there were all of the numbers from 1 to 6. Immediately Stirling reflected: what is the probability of such an event? What is the probability that when throwing 10 dice each number of points from 1 to 6 will drop out on at least one die?
According to one implausible legend, Cauchy and Bunyakovsky were very fond of playing darts in the evenings. But the target was unusual – the sectors on it were unequal, so the probability of getting into different sectors was not the same. Once Cauchy throws a dart and hits the target. Bunyakovsky throws the next one. Which is more likely: that Bunyakovsky will hit the same sector that Cauchy’s dart went into, or that his dart will land on the next sector clockwise?
Four outwardly identical coins weigh 1, 2, 3 and 4 grams respectively.
Is it possible to find out in four weighings on a set of scales without weights, which one weighs how much?
In each cell of a board of size \(5\times5\) a cross or a nought is placed, and no three crosses are positioned in a row, either horizontally, vertically or diagonally. What is the largest number of crosses on the board?
An after school club is attended by 4 boys from class 7A, and four from class 7B. Of those who attended three were named Ben, three were named Will, and two were named Tom.
Is it possible for it to be the case that each boy had at least one namesake classmate who attended the club?
Authors: B. Vysokanov, N. Medved, V. Bragin
The teacher grades tests on a scale from 0 to 100. The school can change the upper bound of the scale to any other natural number, recalculating the estimates proportionally and rounding up to integers. A non-integer number, when rounded, changes to the nearest integer; if the fractional part is equal to 0.5, the direction of rounding can be either up or down and it can be different for each question. (For example, an estimate of 37 on a scale of 100 after recalculation in the scale of 40 will go to \(37 \cdot 40/100 = 14.8\) and will be rounded to 15).
The students of Peter and Valerie got marks, which are not 0 and 100. Prove that the school can do several conversions so that Peter’s mark becomes b and Valerie’s mark becomes a (both marks are recalculated simultaneously).
Three cyclists travel in one direction along a circular track that is 300 meters long. Each of them moves with a constant speed, with all of their speeds being different. A photographer will be able to make a successful photograph of the cyclists, if all of them are on some part of the track which has a length of \(d\) meters. What is the smallest value of \(d\) for which the photographer will be able to make a successful photograph sooner or later?
We took several positive numbers and constructed the following sequence: \(a_1\) is the sum of the initial numbers, \(a_2\) is the sum of the squares of the original numbers, \(a_3\) is the sum of the cubes of the original numbers, and so on.
a) Could it happen that up to \(a_5\) the sequence decreases (\(a_1> a_2> a_3> a_4> a_5\)), and starting with \(a_5\) – it increases (\(a_5 < a_6 < a_7 <\dots\))?
b) Could it be the other way around: before \(a_5\) the sequence increases, and starting with \(a_5\) – decreases?