George drew an empty table of size \(50 \times 50\) and wrote on top of each column and to the left of each row, a number. It turned out that all 100 written numbers are different, and 50 of them are rational, and the remaining 50 are irrational. Then, in each cell of the table, he wrote down the sum of the numbers written at the start of the corresponding row and column (“addition table”). What is the largest number of sums in this table that could be rational numbers?
Gary drew an empty table of \(50 \times 50\) and wrote on top of each column and to the left of each row a number. It turned out that all 100 written numbers are different, and 50 of them are rational, and the remaining 50 are irrational. Then, in each cell of the table, he wrote down a product of numbers written at the top of its column and to the left of the row (the “multiplication table”). What is the largest number of products in this table which could be rational numbers?
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 \(n\)?
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.
Author: L.N. Vaserstein
For any natural numbers \(a_1, a_2, \dots , a_m\), no two of which are equal to each other and none of which is divisible by the square of a natural number greater than one, and also for any integers and non-zero integers \(b_1, b_2, \dots , b_m\) the sum is not zero. Prove this.
Author: V.A. Popov
On the interval \([0; 1]\) a function \(f\) is given. This function is non-negative at all points, \(f (1) = 1\) and, finally, for any two non-negative numbers \(x_1\) and \(x_2\) whose sum does not exceed 1, the quantity \(f (x_1 + x_2)\) does not exceed the sum of \(f (x_1)\) and \(f (x_2)\).
a) Prove that for any number \(x\) on the interval \([0; 1]\), the inequality \(f (x_2) \leq 2x\) holds.
b) Prove that for any number \(x\) on the interval \([0; 1]\), the \(f (x_2) \leq 1.9x\) must be true?
The triangle \(C_1C_2O\) is given. Within it the bisector \(C_2C_3\) is drawn, then in the triangle \(C_2C_3O\) – bisector \(C_3C_4\) and so on. Prove that the sequence of angles \(\gamma_n = C_{n + 1}C_nO\) tends to a limit, and find this limit if \(C_1OC_2 = \alpha\).
A rectangular chocolate bar size \(5 \times 10\) is divided by vertical and horizontal division lines into 50 square pieces. Two players are playing the following game. The one who starts breaks the chocolate bar along some division line into two rectangular pieces and puts the resulting pieces on the table. Then players take turns doing the same operation: each time the player whose turn it is at the moment breaks one of the parts into two parts. The one who is the first to break off a square slice \(1\times 1\) (without division lines) a) loses; b) wins. Which of the players can secure a win: the one who starts or the other one?
What has a greater value: \(300!\) or \(100^{300}\)?
Numbers \(1, 2, 3, \dots , 101\) are written out in a row in some order. Prove that one can cross out 90 of them so that the remaining 11 will be arranged in their magnitude (either increasing or decreasing).