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?
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 \(n\)?
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 \(a + b + \sqrt{ab} > c\).
Suppose that in each issue of our journal in the “Quantum” problem book there are five mathematics problems. We denote by \(f (x, y)\) the number of the first of the problems of the \(x\)-th issue for the \(y\)-th year. Write a general formula for \(f (x, y)\), where \(1 \geq x \geq 12\) and \(1970 \geq y \geq 1989\). Solve the equation \(f (x, y) = y\). For example, \(f (6, 1970) = 26\). Since \(1989\), the number of tasks has become less predictable. For example, in recent years, half the issues have 5 tasks, and in other issues there are 10. Even the number of magazine issues has changed, no longer being 12 but now 6.
A sequence of natural numbers \(a_1 < a_2 < a_3 < \dots < a_n < \dots\) is such that each natural number is either a term in the sequence, can be expressed as the sum of two terms in the sequence, or perhaps the same term twice. Prove that \(a_n \leq n^2\) for any \(n=1, 2, 3,\dots\)
Let’s call a natural number good if in its decimal record we have the numbers 1, 9, 7, 3 in succession, and bad if otherwise. (For example, the number 197,639,917 is bad and the number 116,519,732 is good.) Prove that there exists a positive integer \(n\) such that among all \(n\)-digit numbers (from \(10^{n-1}\) to \(10^{n-1}\)) there are more good than bad numbers.
Try to find the smallest possible \(n\).