We solve problems

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.

Problem #PRU-73620

Problems Calculus Real numbers Integer and fractional parts. Archimedean property Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

13–15 3.0

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.

Problem #PRU-73789

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

Author: A.A. Egorov

Calculate the square root of the number $0.111 \dots 111$ (100 ones) to within a) 100; b) 101; c)* 200 decimal places.

Problem #PRU-73808

Problems Methods Examples and counterexamples. Constructive proofs Algebra Functional equations Calculus Functions of one variable. Continuity Monotonicity, boundedness

14–15 4.0

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 $f (x_{2}) \leq 2 x$ holds.

b) Prove that for any number $x$ on the interval $[0; 1]$ , the $f (x_{2}) \leq 1.9 x$ must be true?

Problem #PRU-73828

Problems Methods Examples and counterexamples. Constructive proofs Pigeonhole principle Pigeonhole principle (other) Set theory and logic Theory of algotithms Theory of algorithms (other)

13–16 4.0

The triangle $C_{1} C_{2} O$ is given. Within it the bisector $C_{2} C_{3}$ is drawn, then in the triangle $C_{2} C_{3} O$ – bisector $C_{3} C_{4}$ and so on. Prove that the sequence of angles $γ_{n} = C_{n + 1} C_{n} O$ tends to a limit, and find this limit if $C_{1} O C_{2} = α$ .

Problem #PRU-74569

Problems Set theory and logic Theory of algotithms Game theory Game theory (other)

12–15 4.0

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?

Problem #PRU-76475

Problems Algebra and arithmetic Number theory. Divisibility Division with remainders. Arithmetic of remainders Division with remainder Calculus Functions of one variable. Continuity Periodicity and aperiodicity Probability and statistics Probability theory Probability theory (other) Methods Algebraic methods Proof by exhaustion

13–16 4.0

What has a greater value: $300!$ or $100^{300}$ ?

Problem #PRU-77892

Problems Combinatorics Painting problems Methods Pigeonhole principle Pigeonhole principle (other) Proof by contradiction Algebra and arithmetic Word problems Tables and tournaments Tables and tournaments (other)

12–14 3.0

There are 13 weights, each weighing an integer number of grams. It is known that any 12 of them can be divided into two cups of weights, six weights on each one, which will come to equilibrium. Prove that all the weights have the same weight.

Problem #PRU-77915

Problems Methods Pigeonhole principle

13–15 3.0

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).

Problem #PRU-77989

Problems Methods Examples and counterexamples. Constructive proofs Algebra and arithmetic Polynomials Quadratic polynomials Quadratic equations and systems of equations

13–16 4.0

The equations $\begin{matrix} (1) & a x^{2} + b x + c = 0 \end{matrix}$ and $\begin{matrix} (2) & - a x^{2} + b x + c \end{matrix}$ are given. Prove that if $x_{1}$ and $x_{2}$ are, respectively, any roots of the equations (1) and (2), then there is a root $x_{3}$ of the equation $\frac{1}{2} a x^{2} + b x + c$ such that either $x_{1} \leq x_{3} \leq x_{2}$ or $x_{1} \geq x_{3} \geq x_{2}$ .