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a$)$ Give an example of a positive number a such that {a} + {1 / a} = 1.

$\\$

b$)$ Can such an a be a rational number?

f$(x)$ is an increasing function defined on the interval [0, 1]. It is known that the range of its values belongs to the interval [0, 1]. Prove that, for any natural N, the graph of the function can be covered by N rectangles whose sides are parallel to the coordinate axes so that the area of each is $1/N^2$. $($In a rectangle we include its interior points and the points of its boundary$)$.

At the cat show, 10 male cats and 19 female cats sit in a row where next to each female cat sits a fatter male cat. Prove that next to each male cat is a female cat, which is thinner than it.

We consider a sequence of words consisting of the letters “A” and “B”. The first word in the sequence is “A”, the k-th word is obtained from the $(k-1)$-th by the following operation: each “A” is replaced by “AAB” and each “B” by “A”. It is easy to see that each word is the beginning of the next, thus obtaining an infinite sequence of letters: AABAABAAABAABAAAB …

$\\$ a) Where in this sequence will the 1000th letter “A” be?

$\\$ b) Prove that this sequence is non-periodic.

The numbers a and b are such that the first equation of the system

$cos x = ax + b$

$sin x + a = 0$

has exactly two solutions. Prove that the system has at least one solution.

The numbers a and b are such that the first equation of the system

$sin x + a = bx$

$cos x = b$

has exactly two solutions. Prove that the system has at least one solution.

Aladdin visited all of the points on the equator, moving to the east, then to the west, and sometimes instantly moving to the diametrically opposite point on Earth. Prove that there was a period of time during which the difference in distances traversed by Aladdin to the east and to the west was not less than half the length of the equator.

Is there a line on the coordinate plane relative to which the graph of the function $y = 2^x$ is symmetric?

Author: A.K. Tolpygo

12 grasshoppers sit on a circle at various points. These points divide the circle into 12 arcs. Let’s mark the 12 mid-points of the arcs. At the signal the grasshoppers jump simultaneously, each to the nearest clockwise marked point. 12 arcs are formed again, and jumps to the middle of the arcs are repeated, etc. Can at least one grasshopper return to his starting point after he has made a) 12 jumps; b) 13 jumps?

Numbers 1, 2, 3, …, 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$)$.

Author: L.N. Vaserstein

For any natural numbers $a_1, a_2, …, a_m$, 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 $b_1, b_2, …, b_m$ the sum is not zero. Prove this.

Find the general formula for the coefficients of the series

$(1 – 4x)^{ ½} = 1 + 2x + 6x^2 + 20x^3 + … + a_nx^n + …$

The Abel transformation. To calculate the integrals, we use the integration by parts formula. Prove the following two formulas, which are a discrete analog of integration by parts and are called the Abel transformation:

f $(x)$ g $(x)$ = f $(n)$

g $(x)$ –

$($ $\Delta f (x)$

g $(z$)$)$,

f $(x)$

$ \Delta g(x) = f (n) g (n) – f (0) g (0)$ –

g $(x + 1)$

$\Delta$ f $(x)$

Let p and q be positive numbers, where 1 / p + 1 / q = 1. Prove that

The values of the variables are considered positive.

Prove that for any $x_1$, …, $x_n$ $\in$ [0; $\pi$] the following inequality holds:

sin $(\frac{x_1+…+x_n}{n}) \geq \frac{sinx_1+…+sinx_n}{n}$

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: $ab ≤ ap/p + bq/q$.

The sequence of numbers $a_1, a_2, a_3$, … 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 \limits_ {n \to \infty} \frac {a_n} {\sqrt [3] n}$.

Old calculator I.

a$)$ Suppose that we want to find $\sqrt[3]{x}$ $(x> 0)$ on a calculator that can find $\sqrt{x}$ in addition to four ordinary arithmetic operations. Consider the following algorithm. A sequence of numbers {$y_n$} is constructed, in which $y_0$ is an arbitrary positive number, for example, $y_0$ = $\sqrt{\sqrt{x}}$, and the remaining elements are defined by

$y_{n + 1}$ = $\sqrt{\sqrt{x y_n}}$ $($n $\geq$ 0$)$.

Prove that

$\lim\limits_{n\to\infty}$ $y_n$ = $\sqrt[3]{x}$.

b$)$ Construct a similar algorithm to calculate the fifth root.

For what values of n does the polynomial $(x+1)^n$ – $x^n$ – 1 divide by:

a$)$ $x^2$ + x + 1; b$)$ $(x^2 + x + 1)^2$; c$)$ $(x^2 + x + 1)^3$?

Let a, b be positive integers and $(a, b)$ = 1. Prove that the quantity cannot be a real number except in the following cases

$4(a, b) = (1, 1), (1,3), (3,1)$.