A sequence consists of 19 ones and 49 zeros, arranged in a random order. We call the maximal subsequence of the same symbols a “group”. For example, in the sequence 110001001111 there are five groups: two ones, then three zeros, then one one, then two zeros and finally four ones. Find the mathematical expectation of the length of the first group.
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 \(|a|\geq \frac{1}{3}\).
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?
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?
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?
The number \(x\) is such that both the sums \(S = \sin 64x + \sin 65x\) and \(C = \cos 64x + \cos 65x\) are rational numbers.
Prove that in both of these sums, both terms are rational.
Author: A.K. Tolpygo
An irrational number \(\alpha\), where \(0 <\alpha <\frac 12\), is given. It defines a new number \(\alpha_1\) as the smaller of the two numbers \(2\alpha\) and \(1 - 2\alpha\). For this number, \(\alpha_2\) is determined similarly, and so on.
a) Prove that for some \(n\) the inequality \(\alpha_n <3/16\) holds.
b) Can it be that \(\alpha_n> 7/40\) for all positive integers \(n\)?
There was a football match of 10 versus 10 players between a team of liars (who always lie) and a team of truth-tellers (who always tell the truth). After the match, each player was asked: “How many goals did you score?” Some participants answered “one”, Callum said “two”, some answered “three”, and the rest said “five”. Is Callum lying if it is known that the truth-tellers won with a score of 20:17?
For all real \(x\) and \(y\), the equality \(f (x^2 + y) = f (x) + f (y^2)\) holds. Find \(f(-1)\).
It is known that \(a = x+y + \sqrt{xy}\), \(b = y + z + \sqrt{yz}\), \(c = x + z + \sqrt{xz}\). where \(x > 0\), \(y > 0\), \(z > 0\). Prove that \(a + b + \sqrt{ab} > c\).