We solve problems

Problem #PRU-109704

Problems Algebra Algebraic inequalities and systems of inequalities Algebraic inequalities (other) Methods Algebraic methods Counting in two ways Calculus Real numbers Integer and fractional parts. Archimedean property

13–15 4.0

Prove that for any positive integer \(n\) the inequality

is true.

Problem #PRU-116021

Problems Algebra Algebraic inequalities and systems of inequalities Number inequalities. Number comparison

13–15 2.0

Find the largest natural number \(n\) which satisfies \(n^{200} <5^{300}\).

Problem #PRU-116145

Problems Algebra Algebraic inequalities and systems of inequalities Linear inequalites and systems of inequalities Set theory and logic Logic and set theory

12–14 2.0

Of the four inequalities \(2x > 70\), \(x < 100\), \(4x > 25\) and \(x > 5\), two are true and two are false. Find the value of \(x\) if it is known that it is an integer.

Problem #PRU-32032

Problems Algebra Algebraic inequalities and systems of inequalities Linear inequalites and systems of inequalities Methods Pigeonhole principle Pigeonhole principle (angles and lengths) Geometry Plane geometry Geometrical inequalities Triangle inequality Triangle inequality (other)

13–14 3.0

On a plane, there are 1983 points and a circle of unit radius. Prove that there is a point on the circle, from which the sum of the distances to these points is no less than 1983.

Problem #PRU-61353

Problems Algebra Algebraic inequalities and systems of inequalities Classical inequalities

11–14 2.0

Prove that \(\sqrt{\frac{a^2 + b^2}{2}} \geq \frac{a+b}{2}\).

Problem #PRU-61392

Problems Algebra Algebraic inequalities and systems of inequalities Classical inequalities

13–15 3.0

We are given rational positive numbers \(p, q\) where \(1/p + 1/q = 1\). Prove that for positive \(a\) and \(b\), the following inequality holds: \(ab \leq \frac{a^p}{p} + \frac{b^q}{q}\).

Problem #PRU-61411

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

13–15 3.0

Let \(p\) and \(q\) be positive numbers where \(1 / p + 1 / q = 1\). Prove that \[a_1b_1 + a_2b_2 + \dots + a_nb_n \leq (a_1^p + \dots a_n^p)^{1/p}(b_1^q +\dots + b_n^q)^{1/q}\] The values of the variables are considered positive.

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.