We solve problems

Problem #PRU-109685

Problems Algebra Algebraic inequalities and systems of inequalities Linear inequalites and systems of inequalities Methods Proof by contradiction Calculus Real numbers Rational and irrational numbers

14–16 4.0

At all rational points of the real line, integers are arranged. Prove that there is a segment such that the sum of the numbers at its ends does not exceed twice the number on its middle.

Problem #PRU-110122

Problems Algebra Algebraic inequalities and systems of inequalities Algebraic inequalities (other) Calculus Functions of one variable. Continuity Monotonicity, boundedness

14–16 4.0

The functions \(f (x) - x\) and \(f (x^2) - x^6\) are defined for all positive \(x\) and increase. Prove that the function

also increases for all positive \(x\).

Problem #PRU-115512

Problems Algebra Algebraic inequalities and systems of inequalities Algebraic inequalities (other)

15–16 4.0

Prove that if the numbers \(x, y, z\) satisfy the following system of equations for some values of \(p\) and \(q\): \[\begin{aligned} y &= x^2 + px + q,\\ z &= y^2 + py + q,\\ x &= z^2 + pz + q, \end{aligned}\] then the inequality \(x^2y + y^2z + z^2x \geq x^2z + y^2x + z^2y\) is satisfied.

Problem #PRU-116627

Problems Algebra Algebraic inequalities and systems of inequalities Algebraic inequalities (other) Polynomials Algebraic identities for polynomials Factoring polynomials Calculus Real numbers Integer and fractional parts. Archimedean property

14–16 3.0

Solve the inequality: \(\lfloor x\rfloor \times \{x\} < x - 1\).

Problem #PRU-35052

Problems Algebra Algebraic inequalities and systems of inequalities Classical inequalities Probability and statistics Probability theory Discrete distribution Number inequalities. Number comparison

14–16 4.0

There are 30 students in the class. Prove that the probability that some two students have the same birthday is more than 50%.

Problem #PRU-60846

Problems Algebra Algebraic inequalities and systems of inequalities Linear inequalites and systems of inequalities Calculus Number sequences Number sequences (other)

15–16 3.0

Prove that in any infinite decimal fraction you can rearrange the numbers so that the resulting fraction becomes a rational number.

Problem #PRU-64548

Problems Calculus Functions of one variable. Continuity Monotonicity, boundedness Algebra Algebraic inequalities and systems of inequalities Quadratic inequalites (several variables)

3.0

Find the largest value of the expression \(a + b + c + d - ab - bc - cd - da\), if each of the numbers \(a\), \(b\), \(c\) and \(d\) belongs to the interval \([0, 1]\).

Problem #PRU-65263

Problems Probability and statistics Probability theory Discrete distribution Algebra Algebraic inequalities and systems of inequalities Quadratic inequalites (several variables) Algebraic equations and systems of equations Systems of algebraic non-linear equations

13–16 4.0

Is it possible to:

a) load two coins so that the probability of “heads” and “tails” were different, and the probability of getting any of the combinations “tails, tails,” “heads, tails”, “heads, heads” be the same?

b) load two dice so that the probability of getting any amount from 2 to 12 would be the same?

Problem #PRU-65296

Problems Probability and statistics Probability theory Conditional probability Algebra Algebraic inequalities and systems of inequalities Linear inequalites and systems of inequalities Sequences Recurrent relations Linear recurrent relations

13–16 3.0

On a calculator keypad, there are the numbers from 0 to 9 and signs of two actions (see the figure). First, the display shows the number 0. You can press any keys. The calculator performs the actions in the sequence of clicks. If the action sign is pressed several times, the calculator will only remember the last click.

a) The button with the multiplier sign breaks and does not work. The Scattered Scientist pressed several buttons in a random sequence. Which result of the resulting sequence of actions is more likely: an even number or an odd number?

b) Solve the previous problem if the multiplication symbol button is repaired.

Problem #PRU-65318

Problems Algebra Polynomials Quadratic polynomials Investigating the quadratic polynomial Mean values Algebraic inequalities and systems of inequalities Quadratic inequalites (several variables) Probability and statistics Statistics

14–16 3.0

In a numerical set of \(n\) numbers, one of the numbers is 0 and another is 1.

a) What is the smallest possible variance of such a set of numbers?

b) What should be the set of numbers for this?