We solve problems

Problem #PRU-61314

Problems Calculus Number sequences Limit of a sequence, convergence Algebra Sequences Recurrent relations Recurrent relations (other)

15–16 4.0

Find the limit of the sequence that is given by the following conditions $a_{1} = 2$ , $a_{n + 1} = a_{n} / 2 + a_{n}^{2} / 8$ ( $n \geq 1$ ).

Problem #PRU-61319

Problems Calculus Number sequences Boundedness, monotonicity Algebra Sequences Recurrent relations Recurrent relations (other)

15–16 4.0

The sequence of numbers ${x_{n}}$ is given by the following conditions: $x_{1} \geq - a$ , $x_{n + 1} = \sqrt{a + x_{n}}$ . Prove that the sequence $x_{n}$ is monotonic and bounded. Find its limit.

Problem #PRU-61320

Problems Methods Algebraic methods Iterations Calculus Functions of one variable. Continuity Monotonicity, boundedness

13–15 2.0

Prove that for a monotonically increasing function $f (x)$ the equations $x = f (f (x))$ and $x = f (x)$ are equivalent.

Problem #PRU-61324

Problems Calculus Number sequences Limit of a sequence, convergence Algebra Mean values Sequences Recurrent relations Recurrent relations (other)

15–16 3.0

We call the geometric-harmonic mean of numbers $a$ and $b$ the general limit of the sequences ${a_{n}}$ and ${b_{n}}$ constructed according to the rule $a_{0} = a$ , $b_{0} = b$ , $a_{n + 1} = \frac{2 a_{n} b_{n}}{a_{n} + b_{n}}$ , $b_{n + 1} = \sqrt{a_{n} b_{n}}$ ( $n \geq 0$ ).

We denote it by $ν (a, b)$ . Prove that $ν (a, b)$ is related to $μ (a, b)$ (see problem number 61322) by $ν (a, b) \times μ (1 / a, 1 / b) = 1$ .

Problem number 61322 says that both of these sequences have the same limit.

This limit is called the arithmetic-geometric mean of the numbers $a, b$ and is denoted by $μ (a, b)$ .

Problem #PRU-61327

Problems Calculus Derivative Derivative and tangent line

15–16 3.0

Prove that the tangent to the graph of the function $f (x)$ , constructed at coordinates $(x_{0}, f (x_{0}))$ intersects the $O x$ axis at the coordinate: $x_{0} - \frac{f (x_{0})}{f^{'} (x_{0})}$ .

Problem #PRU-61332

Problems Calculus Derivative Derivative and tangent line Methods Examples and counterexamples. Constructive proofs

15–16 3.0

The Newton method (see Problem 61328) does not always allow us to approach the root of the equation $f (x) = 0$ . Find the initial condition $x_{0}$ for the polynomial $f (x) = x (x - 1) (x + 1)$ such that $f (x_{0}) \neq x_{0}$ and $x_{2} = x_{0}$ .

Problem #PRU-61338

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

13–15 3.0

The sequence of numbers $a_{1}, a_{2}, a_{3}, \dots$ is given by the following conditions $a_{1} = 1$ , $a_{n + 1} = a_{n} + \frac{1}{a_{n}^{2}}$ ( $n \geq 0$ ).

Prove that

a) this sequence is unbounded;

b) $a_{9000} > 30$ ;

c) find the limit $lim_{n \to \infty} \frac{a_{n}}{\sqrt[3]{n}}$ .

Problem #PRU-61350

Problems Algebra Algebraic equations and systems of equations Discrete Mathematics Set theory and logic Mathematical logic Weightings

15–17 5.0

There are 13 weights. It is known that any 12 of them could be placed in 2 scale cups with 6 weights in each cup in such a way that balance will be held.

Prove the mass of all the weights is the same, if it is known that:

a) the mass of each weight in grams is an integer;

b) the mass of each weight in grams is a rational number;

c) the mass of each weight could be any real (not negative) number.

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: $a b \leq \frac{a^{p}}{p} + \frac{b^{q}}{q}$ .