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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 $γ_n$ = $C_{n + 1}C_nO$ tends to a limit, and find this limit if $C_1OC_2$ = α.

Let p and q be nonzero real numbers and $p^2$ – 4q $>$ 0. Prove that the following sequences converge:

a$)$ $y_0$ = 0, $y_{n + 1} = \frac{q}{p-y_n} (n \geq 0)$;

b$)$ $z_0$ = 0, $z_{n + 1}$ = p – q/$z_n$ $( n \geq 0 )$.

Establish a connection between the limiting values of these sequences y *, z * and the roots of the equation $x^2$ – px + q = 0.

The bank of the Nile was approached by a group of six people: three Bedouins, each with his wife. At the shore is a boat with oars, which can withstand only two people at a time. A Bedouin can not allow his wife to be without him whilst in the company of another man. Can the whole group cross to the other side?

Cut the interval [-1, 1] into black and white segments so that the integrals of any a$)$ linear function; b$)$ a square trinomial in white and black segments are equal.

Old calculator II. The derivative of the function ln x for x = 1 is 1. Hence

$\lim\limits_{x\to0}$

= $\lim\limits_{x\to0}$

= 1.

Use this fact to approximate the natural logarithm of the number N. As in Problem 61302, the standard arithmetic operations and the square root extraction operation are allowed.

a$)$ Using geometric considerations, prove that the base and the side of an isosceles triangle with an angle of 36 $^{\circ}$ at the vertex are incommensurable.

b$)$ Invent a geometric proof of the irrationality of $\sqrt{2}$.

Let f $(x)$ be a polynomial of degree n with roots $α_1$, …, $α_n$. We define the polygon M as the convex hull of the points $α_1$, …, $α_n$ on the complex plane. Prove that the roots of the derivative of this polynomial lie inside the polygon M.

Let f $(x)$ = $(x – a)$ $(x – b)$ $(x – c)$ be a polynomial of the third degree with complex roots a, b, c.

Prove that the roots of the derivative of this polynomial lie inside the triangle with vertices at the points a, b, c.

Using the theorem on rational roots of a polynomial $($ see Problem 61013 $)$, prove that if p / q is rational and cos $(p / q)$ $^{\circ}$ ≠ 0, ± 1, ± 1, then

cos $(p / q)$ $^{\circ}$ is irrational.

Prove that there are infinitely many composite numbers among the numbers [$2^k$ $\sqrt{2}$] $(k = 0, 1, …)$.

Prove that for any infinite continued fraction [$a_0$; $a_1$, …, $a_n$, …] there exists a limit of its suitable fractions – an irrational number α. Explain why if this number α is decomposed into an infinite continued fraction by the algorithm of task 60606, then an infinite continued fraction is obtained.

$x_1$ is the real root of the equation $x^2$ + ax + b = 0, $x_2$ is the real root of the equation $x^2$ – ax – b = 0.

Prove that the equation $x^2$ + 2ax + 2b = 0 has a real root, enclosed between $x_1$ and $x_2$. $($ a and b are real numbers $)$.

With a non-zero number, the following operations are allowed: $x \rightarrow \frac{1+x}{x}, x \rightarrow \frac{1-x}{x}$. Is it true that from every non-zero rational number one can obtain each rational number with the help of a finite number of such operations?

A broken calculator carries out only one operation “asterisk”: $a*b = 1 – a/b$. Prove that using this calculator it is possible to carry out all four arithmetic operations (addition, subtraction, multiplication, division).

There are 8 glasses of water on the table. You are allowed to take any two of the glasses and make them have equal volumes of water (by pouring some water from one glass into the other). Prove that, by using such operations, you can eventually get all the glasses to contain equal volumes of water.

Members of the State parliament formed factions in such a way that for any two factions A and B $($ not necessarily different $)$

– also a faction $($ through the set of all parliament members not included in C is denoted $)$. Prove that for any two factions A and B, A B is also a faction.

A magician with a blindfold gives a spectator five cards with the numbers from 1 to 5 written on them. The spectator hides two cards, and gives the other three to the assistant magician. The assistant indicates to the spectator two of them, and the spectator then calls out the numbers of these cards to the magician $($ in the order in which he wants $)$. After that, the magician guesses the numbers of the cards hidden by the spectator. How can the magician and the assistant make sure that the trick always works?

In 10 boxes there are pencils $($ there are no empty boxes $)$. It is known that in different boxes there is a different number of pencils, and in each box, all pencils are of different colors. Prove that from each box you can choose a pencil so that they will all be of different colors.